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Theorem funcf1 17917

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 2735	. . . 4 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
5		eqid 2735	. . . 4 ⊢ (Hom ‘𝐸) = (Hom ‘𝐸)
6		eqid 2735	. . . 4 ⊢ (Id‘𝐷) = (Id‘𝐷)
7		eqid 2735	. . . 4 ⊢ (Id‘𝐸) = (Id‘𝐸)
8		eqid 2735	. . . 4 ⊢ (comp‘𝐷) = (comp‘𝐷)
9		eqid 2735	. . . 4 ⊢ (comp‘𝐸) = (comp‘𝐸)
10		df-br 5149	. . . . . . 7 ⊢ (𝐹(𝐷 Func 𝐸)𝐺 ↔ ⟨𝐹, 𝐺⟩ ∈ (𝐷 Func 𝐸))
11	1, 10	sylib 218	. . . . . 6 ⊢ (𝜑 → ⟨𝐹, 𝐺⟩ ∈ (𝐷 Func 𝐸))
12		funcrcl 17914	. . . . . 6 ⊢ (⟨𝐹, 𝐺⟩ ∈ (𝐷 Func 𝐸) → (𝐷 ∈ Cat ∧ 𝐸 ∈ Cat))
13	11, 12	syl 17	. . . . 5 ⊢ (𝜑 → (𝐷 ∈ Cat ∧ 𝐸 ∈ Cat))
14	13	simpld 494	. . . 4 ⊢ (𝜑 → 𝐷 ∈ Cat)
15	13	simprd 495	. . . 4 ⊢ (𝜑 → 𝐸 ∈ Cat)
16	2, 3, 4, 5, 6, 7, 8, 9, 14, 15	isfunc 17915	. . 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 232	. 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 1141	1 ⊢ (𝜑 → 𝐹:𝐵⟶𝐶)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1086 = wceq 1537 ∈ wcel 2106 ∀wral 3059 ⟨cop 4637 class class class wbr 5148 × cxp 5687 ⟶wf 6559 ‘cfv 6563 (class class class)co 7431 1^st c1st 8011 2^nd c2nd 8012 ↑_m cmap 8865 Xcixp 8936 Basecbs 17245 Hom chom 17309 compcco 17310 Catccat 17709 Idccid 17710 Func cfunc 17905

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-fv 6571 df-ov 7434 df-oprab 7435 df-mpo 7436 df-map 8867 df-ixp 8937 df-func 17909

This theorem is referenced by: funcsect 17923 funcinv 17924 funciso 17925 funcoppc 17926 cofu1 17935 cofucl 17939 cofuass 17940 cofulid 17941 cofurid 17942 funcres 17947 funcres2 17949 wunfunc 17952 wunfuncOLD 17953 funcres2c 17955 fullpropd 17974 fthsect 17979 fthinv 17980 fthmon 17981 ffthiso 17983 cofull 17988 cofth 17989 fuccocl 18021 fucidcl 18022 fuclid 18023 fucrid 18024 fucass 18025 fucsect 18029 fucinv 18030 invfuc 18031 fuciso 18032 natpropd 18033 fucpropd 18034 catciso 18165 prfval 18255 prfcl 18259 prf1st 18260 prf2nd 18261 1st2ndprf 18262 evlfcllem 18278 evlfcl 18279 curf1cl 18285 curfcl 18289 uncf1 18293 uncf2 18294 curfuncf 18295 uncfcurf 18296 diag1cl 18299 curf2ndf 18304 yon1cl 18320 oyon1cl 18328 yonedalem3a 18331 yonedalem4c 18334 yonedalem3b 18336 yonedalem3 18337 yonedainv 18338 yonffthlem 18339 yoniso 18342 upciclem2 48813 upciclem3 48814 upeu2 48818 fullthinc 48846

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