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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 4188 | . 2 ⊢ (𝐹 ∩ 𝐺) ⊆ 𝐹 | |
2 | funss 6520 | . 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 3909 ⊆ wss 3910 Fun wfun 6490 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2707 |
This theorem depends on definitions: df-bi 206 df-an 397 df-tru 1544 df-ex 1782 df-sb 2068 df-clab 2714 df-cleq 2728 df-clel 2814 df-v 3447 df-in 3917 df-ss 3927 df-br 5106 df-opab 5168 df-rel 5640 df-cnv 5641 df-co 5642 df-fun 6498 |
This theorem is referenced by: (None) |
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