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Theorem funcf1 16349

Description: The object part of a functor is a function on objects. (Contributed by Mario Carneiro, 2-Jan-2017.)

**Hypotheses**
Ref	Expression
funcf1.b	⊢ 𝐵 = (Base‘𝐷)
funcf1.c	⊢ 𝐶 = (Base‘𝐸)
funcf1.f	⊢ (𝜑 → 𝐹(𝐷 Func 𝐸)𝐺)

**Assertion**
Ref	Expression
funcf1	⊢ (𝜑 → 𝐹:𝐵⟶𝐶)

Proof of Theorem funcf1 Dummy variables 𝑚 𝑛 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		funcf1.f	. . 3 ⊢ (𝜑 → 𝐹(𝐷 Func 𝐸)𝐺)
2		funcf1.b	. . . 4 ⊢ 𝐵 = (Base‘𝐷)
3		funcf1.c	. . . 4 ⊢ 𝐶 = (Base‘𝐸)
4		eqid 2610	. . . 4 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
5		eqid 2610	. . . 4 ⊢ (Hom ‘𝐸) = (Hom ‘𝐸)
6		eqid 2610	. . . 4 ⊢ (Id‘𝐷) = (Id‘𝐷)
7		eqid 2610	. . . 4 ⊢ (Id‘𝐸) = (Id‘𝐸)
8		eqid 2610	. . . 4 ⊢ (comp‘𝐷) = (comp‘𝐷)
9		eqid 2610	. . . 4 ⊢ (comp‘𝐸) = (comp‘𝐸)
10		df-br 4584	. . . . . . 7 ⊢ (𝐹(𝐷 Func 𝐸)𝐺 ↔ ⟨𝐹, 𝐺⟩ ∈ (𝐷 Func 𝐸))
11	1, 10	sylib 207	. . . . . 6 ⊢ (𝜑 → ⟨𝐹, 𝐺⟩ ∈ (𝐷 Func 𝐸))
12		funcrcl 16346	. . . . . 6 ⊢ (⟨𝐹, 𝐺⟩ ∈ (𝐷 Func 𝐸) → (𝐷 ∈ Cat ∧ 𝐸 ∈ Cat))
13	11, 12	syl 17	. . . . 5 ⊢ (𝜑 → (𝐷 ∈ Cat ∧ 𝐸 ∈ Cat))
14	13	simpld 474	. . . 4 ⊢ (𝜑 → 𝐷 ∈ Cat)
15	13	simprd 478	. . . 4 ⊢ (𝜑 → 𝐸 ∈ Cat)
16	2, 3, 4, 5, 6, 7, 8, 9, 14, 15	isfunc 16347	. . 3 ⊢ (𝜑 → (𝐹(𝐷 Func 𝐸)𝐺 ↔ (𝐹:𝐵⟶𝐶 ∧ 𝐺 ∈ X𝑧 ∈ (𝐵 × 𝐵)(((𝐹‘(1^st ‘𝑧))(Hom ‘𝐸)(𝐹‘(2^nd ‘𝑧))) ↑_𝑚 ((Hom ‘𝐷)‘𝑧)) ∧ ∀𝑥 ∈ 𝐵 (((𝑥𝐺𝑥)‘((Id‘𝐷)‘𝑥)) = ((Id‘𝐸)‘(𝐹‘𝑥)) ∧ ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ∀𝑚 ∈ (𝑥(Hom ‘𝐷)𝑦)∀𝑛 ∈ (𝑦(Hom ‘𝐷)𝑧)((𝑥𝐺𝑧)‘(𝑛(⟨𝑥, 𝑦⟩(comp‘𝐷)𝑧)𝑚)) = (((𝑦𝐺𝑧)‘𝑛)(⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩(comp‘𝐸)(𝐹‘𝑧))((𝑥𝐺𝑦)‘𝑚))))))
17	1, 16	mpbid 221	. 2 ⊢ (𝜑 → (𝐹:𝐵⟶𝐶 ∧ 𝐺 ∈ X𝑧 ∈ (𝐵 × 𝐵)(((𝐹‘(1^st ‘𝑧))(Hom ‘𝐸)(𝐹‘(2^nd ‘𝑧))) ↑_𝑚 ((Hom ‘𝐷)‘𝑧)) ∧ ∀𝑥 ∈ 𝐵 (((𝑥𝐺𝑥)‘((Id‘𝐷)‘𝑥)) = ((Id‘𝐸)‘(𝐹‘𝑥)) ∧ ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ∀𝑚 ∈ (𝑥(Hom ‘𝐷)𝑦)∀𝑛 ∈ (𝑦(Hom ‘𝐷)𝑧)((𝑥𝐺𝑧)‘(𝑛(⟨𝑥, 𝑦⟩(comp‘𝐷)𝑧)𝑚)) = (((𝑦𝐺𝑧)‘𝑛)(⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩(comp‘𝐸)(𝐹‘𝑧))((𝑥𝐺𝑦)‘𝑚)))))
18	17	simp1d 1066	1 ⊢ (𝜑 → 𝐹:𝐵⟶𝐶)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ∀wral 2896 ⟨cop 4131 class class class wbr 4583 × cxp 5036 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 1^st c1st 7057 2^nd c2nd 7058 ↑_𝑚 cmap 7744 Xcixp 7794 Basecbs 15695 Hom chom 15779 compcco 15780 Catccat 16148 Idccid 16149 Func cfunc 16337

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-map 7746 df-ixp 7795 df-func 16341

This theorem is referenced by: funcsect 16355 funcinv 16356 funciso 16357 funcoppc 16358 cofu1 16367 cofucl 16371 cofuass 16372 cofulid 16373 cofurid 16374 funcres 16379 funcres2 16381 wunfunc 16382 funcres2c 16384 fullpropd 16403 fthsect 16408 fthinv 16409 fthmon 16410 ffthiso 16412 cofull 16417 cofth 16418 fuccocl 16447 fucidcl 16448 fuclid 16449 fucrid 16450 fucass 16451 fucsect 16455 fucinv 16456 invfuc 16457 fuciso 16458 natpropd 16459 fucpropd 16460 catciso 16580 prfval 16662 prfcl 16666 prf1st 16667 prf2nd 16668 1st2ndprf 16669 evlfcllem 16684 evlfcl 16685 curf1cl 16691 curfcl 16695 uncf1 16699 uncf2 16700 curfuncf 16701 uncfcurf 16702 diag1cl 16705 curf2ndf 16710 yon1cl 16726 oyon1cl 16734 yonedalem3a 16737 yonedalem4c 16740 yonedalem3b 16742 yonedalem3 16743 yonedainv 16744 yonffthlem 16745 yoniso 16748

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