Math 2.0

Mathematical logic and foundations Proof

Proof that the image of an intersection lies in the intersection of the images

Builds on Image of a setIntersectionSubset

Proof

We show that each element of the left side is an element of the right side.

  1. Let f ⁣:A→Bf \colon A \to B be a function and let S,T⊆AS, T \subseteq A.
  2. Let y∈f(S∩T)y \in f(S \cap T). By the definition of the image, there is s∈S∩Ts \in S \cap T with f(s)=yf(s) = y.
  3. By the definition of intersection, s∈Ss \in S and s∈Ts \in T.
  4. Since s∈Ss \in S and f(s)=yf(s) = y, the definition of the image gives y∈f(S)y \in f(S).
  5. Since s∈Ts \in T and f(s)=yf(s) = y, the definition of the image gives y∈f(T)y \in f(T).
  6. By steps 4 and 5, y∈f(S)∩f(T)y \in f(S) \cap f(T).
  7. Step 2 took an arbitrary element yy of f(S∩T)f(S \cap T). So f(S∩T)f(S \cap T) is a subset of f(S)∩f(T)f(S) \cap f(T).

Idea

A point of S∩TS \cap T that maps to yy is a witness in SS and a witness in TT at the same time. The reverse inclusion can fail, because yy may come from one point of SS and a different point of TT.

Reading path

To understand this page from the ground up, read these 17 pages in order. The foundations come first.

Show the path as a graph
Element, definition, no evidence levelElementdefinitionProposition, concept, no evidence levelPropositionconceptSet, definition, no evidence levelSetdefinitionConjunction, definition, no evidence levelConjunctiondefinitionImplication, definition, no evidence levelImplicationdefinitionPredicate, definition, no evidence levelPredicatedefinitionOrdered pair, definition, no evidence levelOrdered pairdefinitionIntersection, definition, no evidence levelIntersectiondefinitionExistential quantifier, definition, no evidence levelExistential quantifierdefinitionUniversal quantifier, definition, no evidence levelUniversal quantifierdefinitionCartesian product, definition, no evidence levelCartesian productdefinitionSubset, definition, no evidence levelSubsetdefinitionUniqueness quantifier, definition, no evidence levelUniqueness quantifierdefinitionBinary relation, definition, no evidence levelBinary relationdefinitionFunction, definition, no evidence levelFunctiondefinitionImage of a set, definition, no evidence levelImage of a setdefinitionProof that the image of an intersection lies in the intersection of the images, proof, no evidence levelProof that the image …proof
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Element definition
  2. Proposition concept
  3. Set definition
  4. Conjunction definition
  5. Implication definition
  6. Predicate definition
  7. Ordered pair definition
  8. Intersection definition
  9. Existential quantifier definition
  10. Universal quantifier definition
  11. Cartesian product definition
  12. Subset definition
  13. Uniqueness quantifier definition
  14. Binary relation definition
  15. Function definition
  16. Image of a set definition
  17. Proof that the image of an intersection lies in the intersection of the images proof

Where this leads

1 page builds on this one.

Further reading

  1. Web https://proofwiki.org/wiki/Image_of_Intersection_under_Mapping, Proof 1
About this page
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URIm2://3N7QBXTW9A
Kindproof
WritingW0 stub
MSC03E20
ProvesImage of an intersection
UsesImage of a set, Intersection, Subset
Changed4 November 2026

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