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Mirrors > Home > MPE Home > Th. List > funin | Structured version Visualization version GIF version |
Description: The intersection with a function is a function. Exercise 14(a) of [Enderton] p. 53. (Contributed by NM, 19-Mar-2004.) (Proof shortened by Andrew Salmon, 17-Sep-2011.) |
Ref | Expression |
---|---|
funin | ⊢ (Fun 𝐹 → Fun (𝐹 ∩ 𝐺)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | inss1 4229 | . 2 ⊢ (𝐹 ∩ 𝐺) ⊆ 𝐹 | |
2 | funss 6572 | . 2 ⊢ ((𝐹 ∩ 𝐺) ⊆ 𝐹 → (Fun 𝐹 → Fun (𝐹 ∩ 𝐺))) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ (Fun 𝐹 → Fun (𝐹 ∩ 𝐺)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∩ cin 3946 ⊆ wss 3947 Fun wfun 6542 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-ext 2699 |
This theorem depends on definitions: df-bi 206 df-an 396 df-tru 1537 df-ex 1775 df-sb 2061 df-clab 2706 df-cleq 2720 df-clel 2806 df-v 3473 df-in 3954 df-ss 3964 df-br 5149 df-opab 5211 df-rel 5685 df-cnv 5686 df-co 5687 df-fun 6550 |
This theorem is referenced by: (None) |
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