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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 4189 | . 2 ⊢ (𝐹 ∩ 𝐺) ⊆ 𝐹 | |
2 | funss 6521 | . 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 3910 ⊆ wss 3911 Fun wfun 6491 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 |
This theorem depends on definitions: df-bi 206 df-an 398 df-tru 1545 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-v 3446 df-in 3918 df-ss 3928 df-br 5107 df-opab 5169 df-rel 5641 df-cnv 5642 df-co 5643 df-fun 6499 |
This theorem is referenced by: (None) |
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