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Mirrors > Home > ILE Home > Th. List > funin | 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 3186 | . 2 ⊢ (𝐹 ∩ 𝐺) ⊆ 𝐹 | |
2 | funss 4940 | . 2 ⊢ ((𝐹 ∩ 𝐺) ⊆ 𝐹 → (Fun 𝐹 → Fun (𝐹 ∩ 𝐺))) | |
3 | 1, 2 | ax-mp 7 | 1 ⊢ (Fun 𝐹 → Fun (𝐹 ∩ 𝐺)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∩ cin 2972 ⊆ wss 2973 Fun wfun 4916 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 |
This theorem depends on definitions: df-bi 115 df-tru 1287 df-nf 1390 df-sb 1686 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-v 2603 df-in 2979 df-ss 2986 df-br 3786 df-opab 3840 df-rel 4370 df-cnv 4371 df-co 4372 df-fun 4924 |
This theorem is referenced by: (None) |
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