funcf1 - Metamath Proof Explorer

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Theorem funcf1 16747

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 2760	. . . 4 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
5		eqid 2760	. . . 4 ⊢ (Hom ‘𝐸) = (Hom ‘𝐸)
6		eqid 2760	. . . 4 ⊢ (Id‘𝐷) = (Id‘𝐷)
7		eqid 2760	. . . 4 ⊢ (Id‘𝐸) = (Id‘𝐸)
8		eqid 2760	. . . 4 ⊢ (comp‘𝐷) = (comp‘𝐷)
9		eqid 2760	. . . 4 ⊢ (comp‘𝐸) = (comp‘𝐸)
10		df-br 4805	. . . . . . 7 ⊢ (𝐹(𝐷 Func 𝐸)𝐺 ↔ ⟨𝐹, 𝐺⟩ ∈ (𝐷 Func 𝐸))
11	1, 10	sylib 208	. . . . . 6 ⊢ (𝜑 → ⟨𝐹, 𝐺⟩ ∈ (𝐷 Func 𝐸))
12		funcrcl 16744	. . . . . 6 ⊢ (⟨𝐹, 𝐺⟩ ∈ (𝐷 Func 𝐸) → (𝐷 ∈ Cat ∧ 𝐸 ∈ Cat))
13	11, 12	syl 17	. . . . 5 ⊢ (𝜑 → (𝐷 ∈ Cat ∧ 𝐸 ∈ Cat))
14	13	simpld 477	. . . 4 ⊢ (𝜑 → 𝐷 ∈ Cat)
15	13	simprd 482	. . . 4 ⊢ (𝜑 → 𝐸 ∈ Cat)
16	2, 3, 4, 5, 6, 7, 8, 9, 14, 15	isfunc 16745	. . 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 222	. 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 1137	1 ⊢ (𝜑 → 𝐹:𝐵⟶𝐶)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 383 ∧ w3a 1072 = wceq 1632 ∈ wcel 2139 ∀wral 3050 ⟨cop 4327 class class class wbr 4804 × cxp 5264 ⟶wf 6045 ‘cfv 6049 (class class class)co 6814 1^st c1st 7332 2^nd c2nd 7333 ↑_𝑚 cmap 8025 Xcixp 8076 Basecbs 16079 Hom chom 16174 compcco 16175 Catccat 16546 Idccid 16547 Func cfunc 16735

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-8 2141 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 ax-rep 4923 ax-sep 4933 ax-nul 4941 ax-pow 4992 ax-pr 5055 ax-un 7115

This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3an 1074 df-tru 1635 df-ex 1854 df-nf 1859 df-sb 2047 df-eu 2611 df-mo 2612 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-ne 2933 df-ral 3055 df-rex 3056 df-reu 3057 df-rab 3059 df-v 3342 df-sbc 3577 df-csb 3675 df-dif 3718 df-un 3720 df-in 3722 df-ss 3729 df-nul 4059 df-if 4231 df-pw 4304 df-sn 4322 df-pr 4324 df-op 4328 df-uni 4589 df-iun 4674 df-br 4805 df-opab 4865 df-mpt 4882 df-id 5174 df-xp 5272 df-rel 5273 df-cnv 5274 df-co 5275 df-dm 5276 df-rn 5277 df-res 5278 df-ima 5279 df-iota 6012 df-fun 6051 df-fn 6052 df-f 6053 df-f1 6054 df-fo 6055 df-f1o 6056 df-fv 6057 df-ov 6817 df-oprab 6818 df-mpt2 6819 df-map 8027 df-ixp 8077 df-func 16739

This theorem is referenced by: funcsect 16753 funcinv 16754 funciso 16755 funcoppc 16756 cofu1 16765 cofucl 16769 cofuass 16770 cofulid 16771 cofurid 16772 funcres 16777 funcres2 16779 wunfunc 16780 funcres2c 16782 fullpropd 16801 fthsect 16806 fthinv 16807 fthmon 16808 ffthiso 16810 cofull 16815 cofth 16816 fuccocl 16845 fucidcl 16846 fuclid 16847 fucrid 16848 fucass 16849 fucsect 16853 fucinv 16854 invfuc 16855 fuciso 16856 natpropd 16857 fucpropd 16858 catciso 16978 prfval 17060 prfcl 17064 prf1st 17065 prf2nd 17066 1st2ndprf 17067 evlfcllem 17082 evlfcl 17083 curf1cl 17089 curfcl 17093 uncf1 17097 uncf2 17098 curfuncf 17099 uncfcurf 17100 diag1cl 17103 curf2ndf 17108 yon1cl 17124 oyon1cl 17132 yonedalem3a 17135 yonedalem4c 17138 yonedalem3b 17140 yonedalem3 17141 yonedainv 17142 yonffthlem 17143 yoniso 17146

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