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Mirrors > Home > MPE Home > Th. List > funcfn2 | Structured version Visualization version GIF version |
Description: The morphism part of a functor is a function. (Contributed by Mario Carneiro, 3-Jan-2017.) |
Ref | Expression |
---|---|
funcfn2.b | ⊢ 𝐵 = (Base‘𝐷) |
funcfn2.f | ⊢ (𝜑 → 𝐹(𝐷 Func 𝐸)𝐺) |
Ref | Expression |
---|---|
funcfn2 | ⊢ (𝜑 → 𝐺 Fn (𝐵 × 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | funcfn2.b | . . 3 ⊢ 𝐵 = (Base‘𝐷) | |
2 | eqid 2798 | . . 3 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷) | |
3 | eqid 2798 | . . 3 ⊢ (Hom ‘𝐸) = (Hom ‘𝐸) | |
4 | funcfn2.f | . . 3 ⊢ (𝜑 → 𝐹(𝐷 Func 𝐸)𝐺) | |
5 | 1, 2, 3, 4 | funcixp 17129 | . 2 ⊢ (𝜑 → 𝐺 ∈ X𝑥 ∈ (𝐵 × 𝐵)(((𝐹‘(1st ‘𝑥))(Hom ‘𝐸)(𝐹‘(2nd ‘𝑥))) ↑m ((Hom ‘𝐷)‘𝑥))) |
6 | ixpfn 8450 | . 2 ⊢ (𝐺 ∈ X𝑥 ∈ (𝐵 × 𝐵)(((𝐹‘(1st ‘𝑥))(Hom ‘𝐸)(𝐹‘(2nd ‘𝑥))) ↑m ((Hom ‘𝐷)‘𝑥)) → 𝐺 Fn (𝐵 × 𝐵)) | |
7 | 5, 6 | syl 17 | 1 ⊢ (𝜑 → 𝐺 Fn (𝐵 × 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1538 ∈ wcel 2111 class class class wbr 5030 × cxp 5517 Fn wfn 6319 ‘cfv 6324 (class class class)co 7135 1st c1st 7669 2nd c2nd 7670 ↑m cmap 8389 Xcixp 8444 Basecbs 16475 Hom chom 16568 Func cfunc 17116 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-ov 7138 df-oprab 7139 df-mpo 7140 df-map 8391 df-ixp 8445 df-func 17120 |
This theorem is referenced by: funcoppc 17137 cofuval 17144 cofulid 17152 cofurid 17153 prf1st 17446 prf2nd 17447 1st2ndprf 17448 curfuncf 17480 uncfcurf 17481 curf2ndf 17489 |
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