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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 2823 | . . 3 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷) | |
3 | eqid 2823 | . . 3 ⊢ (Hom ‘𝐸) = (Hom ‘𝐸) | |
4 | funcfn2.f | . . 3 ⊢ (𝜑 → 𝐹(𝐷 Func 𝐸)𝐺) | |
5 | 1, 2, 3, 4 | funcixp 17139 | . 2 ⊢ (𝜑 → 𝐺 ∈ X𝑥 ∈ (𝐵 × 𝐵)(((𝐹‘(1st ‘𝑥))(Hom ‘𝐸)(𝐹‘(2nd ‘𝑥))) ↑m ((Hom ‘𝐷)‘𝑥))) |
6 | ixpfn 8469 | . 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 1537 ∈ wcel 2114 class class class wbr 5068 × cxp 5555 Fn wfn 6352 ‘cfv 6357 (class class class)co 7158 1st c1st 7689 2nd c2nd 7690 ↑m cmap 8408 Xcixp 8463 Basecbs 16485 Hom chom 16578 Func cfunc 17126 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-reu 3147 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-ov 7161 df-oprab 7162 df-mpo 7163 df-map 8410 df-ixp 8464 df-func 17130 |
This theorem is referenced by: funcoppc 17147 cofuval 17154 cofulid 17162 cofurid 17163 prf1st 17456 prf2nd 17457 1st2ndprf 17458 curfuncf 17490 uncfcurf 17491 curf2ndf 17499 |
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