funcf1 - Metamath Proof Explorer

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Theorem funcf1 16994

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 2778	. . . 4 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
5		eqid 2778	. . . 4 ⊢ (Hom ‘𝐸) = (Hom ‘𝐸)
6		eqid 2778	. . . 4 ⊢ (Id‘𝐷) = (Id‘𝐷)
7		eqid 2778	. . . 4 ⊢ (Id‘𝐸) = (Id‘𝐸)
8		eqid 2778	. . . 4 ⊢ (comp‘𝐷) = (comp‘𝐷)
9		eqid 2778	. . . 4 ⊢ (comp‘𝐸) = (comp‘𝐸)
10		df-br 4930	. . . . . . 7 ⊢ (𝐹(𝐷 Func 𝐸)𝐺 ↔ ⟨𝐹, 𝐺⟩ ∈ (𝐷 Func 𝐸))
11	1, 10	sylib 210	. . . . . 6 ⊢ (𝜑 → ⟨𝐹, 𝐺⟩ ∈ (𝐷 Func 𝐸))
12		funcrcl 16991	. . . . . 6 ⊢ (⟨𝐹, 𝐺⟩ ∈ (𝐷 Func 𝐸) → (𝐷 ∈ Cat ∧ 𝐸 ∈ Cat))
13	11, 12	syl 17	. . . . 5 ⊢ (𝜑 → (𝐷 ∈ Cat ∧ 𝐸 ∈ Cat))
14	13	simpld 487	. . . 4 ⊢ (𝜑 → 𝐷 ∈ Cat)
15	13	simprd 488	. . . 4 ⊢ (𝜑 → 𝐸 ∈ Cat)
16	2, 3, 4, 5, 6, 7, 8, 9, 14, 15	isfunc 16992	. . 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 224	. 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 1122	1 ⊢ (𝜑 → 𝐹:𝐵⟶𝐶)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 387 ∧ w3a 1068 = wceq 1507 ∈ wcel 2050 ∀wral 3088 ⟨cop 4447 class class class wbr 4929 × cxp 5405 ⟶wf 6184 ‘cfv 6188 (class class class)co 6976 1^st c1st 7499 2^nd c2nd 7500 ↑_𝑚 cmap 8206 Xcixp 8259 Basecbs 16339 Hom chom 16432 compcco 16433 Catccat 16793 Idccid 16794 Func cfunc 16982

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2750 ax-rep 5049 ax-sep 5060 ax-nul 5067 ax-pow 5119 ax-pr 5186 ax-un 7279

This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2584 df-clab 2759 df-cleq 2771 df-clel 2846 df-nfc 2918 df-ne 2968 df-ral 3093 df-rex 3094 df-reu 3095 df-rab 3097 df-v 3417 df-sbc 3682 df-csb 3787 df-dif 3832 df-un 3834 df-in 3836 df-ss 3843 df-nul 4179 df-if 4351 df-pw 4424 df-sn 4442 df-pr 4444 df-op 4448 df-uni 4713 df-iun 4794 df-br 4930 df-opab 4992 df-mpt 5009 df-id 5312 df-xp 5413 df-rel 5414 df-cnv 5415 df-co 5416 df-dm 5417 df-rn 5418 df-res 5419 df-ima 5420 df-iota 6152 df-fun 6190 df-fn 6191 df-f 6192 df-f1 6193 df-fo 6194 df-f1o 6195 df-fv 6196 df-ov 6979 df-oprab 6980 df-mpo 6981 df-map 8208 df-ixp 8260 df-func 16986

This theorem is referenced by: funcsect 17000 funcinv 17001 funciso 17002 funcoppc 17003 cofu1 17012 cofucl 17016 cofuass 17017 cofulid 17018 cofurid 17019 funcres 17024 funcres2 17026 wunfunc 17027 funcres2c 17029 fullpropd 17048 fthsect 17053 fthinv 17054 fthmon 17055 ffthiso 17057 cofull 17062 cofth 17063 fuccocl 17092 fucidcl 17093 fuclid 17094 fucrid 17095 fucass 17096 fucsect 17100 fucinv 17101 invfuc 17102 fuciso 17103 natpropd 17104 fucpropd 17105 catciso 17225 prfval 17307 prfcl 17311 prf1st 17312 prf2nd 17313 1st2ndprf 17314 evlfcllem 17329 evlfcl 17330 curf1cl 17336 curfcl 17340 uncf1 17344 uncf2 17345 curfuncf 17346 uncfcurf 17347 diag1cl 17350 curf2ndf 17355 yon1cl 17371 oyon1cl 17379 yonedalem3a 17382 yonedalem4c 17385 yonedalem3b 17387 yonedalem3 17388 yonedainv 17389 yonffthlem 17390 yoniso 17393

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