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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 2736 | . . 3 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷) | |
| 3 | eqid 2736 | . . 3 ⊢ (Hom ‘𝐸) = (Hom ‘𝐸) | |
| 4 | funcfn2.f | . . 3 ⊢ (𝜑 → 𝐹(𝐷 Func 𝐸)𝐺) | |
| 5 | 1, 2, 3, 4 | funcixp 17913 | . 2 ⊢ (𝜑 → 𝐺 ∈ X𝑥 ∈ (𝐵 × 𝐵)(((𝐹‘(1st ‘𝑥))(Hom ‘𝐸)(𝐹‘(2nd ‘𝑥))) ↑m ((Hom ‘𝐷)‘𝑥))) | 
| 6 | ixpfn 8944 | . 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 1539 ∈ wcel 2107 class class class wbr 5142 × cxp 5682 Fn wfn 6555 ‘cfv 6560 (class class class)co 7432 1st c1st 8013 2nd c2nd 8014 ↑m cmap 8867 Xcixp 8938 Basecbs 17248 Hom chom 17309 Func cfunc 17900 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-rep 5278 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-iun 4992 df-br 5143 df-opab 5205 df-mpt 5225 df-id 5577 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-fv 6568 df-ov 7435 df-oprab 7436 df-mpo 7437 df-map 8869 df-ixp 8939 df-func 17904 | 
| This theorem is referenced by: funcoppc 17921 cofuval 17928 cofulid 17936 cofurid 17937 prf1st 18250 prf2nd 18251 1st2ndprf 18252 curfuncf 18284 uncfcurf 18285 curf2ndf 18293 diag1 49022 termcfuncval 49190 | 
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