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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 2622 | . . 3 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷) | |
3 | eqid 2622 | . . 3 ⊢ (Hom ‘𝐸) = (Hom ‘𝐸) | |
4 | funcfn2.f | . . 3 ⊢ (𝜑 → 𝐹(𝐷 Func 𝐸)𝐺) | |
5 | 1, 2, 3, 4 | funcixp 16527 | . 2 ⊢ (𝜑 → 𝐺 ∈ X𝑥 ∈ (𝐵 × 𝐵)(((𝐹‘(1st ‘𝑥))(Hom ‘𝐸)(𝐹‘(2nd ‘𝑥))) ↑𝑚 ((Hom ‘𝐷)‘𝑥))) |
6 | ixpfn 7914 | . 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 1483 ∈ wcel 1990 class class class wbr 4653 × cxp 5112 Fn wfn 5883 ‘cfv 5888 (class class class)co 6650 1st c1st 7166 2nd c2nd 7167 ↑𝑚 cmap 7857 Xcixp 7908 Basecbs 15857 Hom chom 15952 Func cfunc 16514 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-map 7859 df-ixp 7909 df-func 16518 |
This theorem is referenced by: funcoppc 16535 cofuval 16542 cofulid 16550 cofurid 16551 prf1st 16844 prf2nd 16845 1st2ndprf 16846 curfuncf 16878 uncfcurf 16879 curf2ndf 16887 |
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