Theorem cofulid 17521

Description: The identity functor is a left identity for composition. (Contributed by Mario Carneiro, 3-Jan-2017.)

Hypotheses

Ref	Expression
cofulid.g	⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))
cofulid.1	⊢ 𝐼 = (idfunc‘𝐷)

Assertion

Ref	Expression
cofulid	⊢ (𝜑 → (𝐼 ∘func 𝐹) = 𝐹)

Proof of Theorem cofulid

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		cofulid.1	. . . . . 6 ⊢ 𝐼 = (idfunc‘𝐷)
2		eqid 2738	. . . . . 6 ⊢ (Base‘𝐷) = (Base‘𝐷)
3		cofulid.g	. . . . . . . 8 ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))
4		funcrcl 17494	. . . . . . . 8 ⊢ (𝐹 ∈ (𝐶 Func 𝐷) → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
5	3, 4	syl 17	. . . . . . 7 ⊢ (𝜑 → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
6	5	simprd 495	. . . . . 6 ⊢ (𝜑 → 𝐷 ∈ Cat)
7	1, 2, 6	idfu1st 17510	. . . . 5 ⊢ (𝜑 → (1st ‘𝐼) = ( I ↾ (Base‘𝐷)))
8	7	coeq1d 5759	. . . 4 ⊢ (𝜑 → ((1st ‘𝐼) ∘ (1st ‘𝐹)) = (( I ↾ (Base‘𝐷)) ∘ (1st ‘𝐹)))
9		eqid 2738	. . . . . 6 ⊢ (Base‘𝐶) = (Base‘𝐶)
10		relfunc 17493	. . . . . . 7 ⊢ Rel (𝐶 Func 𝐷)
11		1st2ndbr 7856	. . . . . . 7 ⊢ ((Rel (𝐶 Func 𝐷) ∧ 𝐹 ∈ (𝐶 Func 𝐷)) → (1st ‘𝐹)(𝐶 Func 𝐷)(2nd ‘𝐹))
12	10, 3, 11	sylancr 586	. . . . . 6 ⊢ (𝜑 → (1st ‘𝐹)(𝐶 Func 𝐷)(2nd ‘𝐹))
13	9, 2, 12	funcf1 17497	. . . . 5 ⊢ (𝜑 → (1st ‘𝐹):(Base‘𝐶)⟶(Base‘𝐷))
14		fcoi2 6633	. . . . 5 ⊢ ((1st ‘𝐹):(Base‘𝐶)⟶(Base‘𝐷) → (( I ↾ (Base‘𝐷)) ∘ (1st ‘𝐹)) = (1st ‘𝐹))
15	13, 14	syl 17	. . . 4 ⊢ (𝜑 → (( I ↾ (Base‘𝐷)) ∘ (1st ‘𝐹)) = (1st ‘𝐹))
16	8, 15	eqtrd 2778	. . 3 ⊢ (𝜑 → ((1st ‘𝐼) ∘ (1st ‘𝐹)) = (1st ‘𝐹))
17	6	3ad2ant1 1131	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → 𝐷 ∈ Cat)
18		eqid 2738	. . . . . . . 8 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
19	13	ffvelrnda 6943	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((1st ‘𝐹)‘𝑥) ∈ (Base‘𝐷))
20	19	3adant3 1130	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → ((1st ‘𝐹)‘𝑥) ∈ (Base‘𝐷))
21	13	ffvelrnda 6943	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (Base‘𝐶)) → ((1st ‘𝐹)‘𝑦) ∈ (Base‘𝐷))
22	21	3adant2 1129	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → ((1st ‘𝐹)‘𝑦) ∈ (Base‘𝐷))
23	1, 2, 17, 18, 20, 22	idfu2nd 17508	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → (((1st ‘𝐹)‘𝑥)(2nd ‘𝐼)((1st ‘𝐹)‘𝑦)) = ( I ↾ (((1st ‘𝐹)‘𝑥)(Hom ‘𝐷)((1st ‘𝐹)‘𝑦))))
24	23	coeq1d 5759	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → ((((1st ‘𝐹)‘𝑥)(2nd ‘𝐼)((1st ‘𝐹)‘𝑦)) ∘ (𝑥(2nd ‘𝐹)𝑦)) = (( I ↾ (((1st ‘𝐹)‘𝑥)(Hom ‘𝐷)((1st ‘𝐹)‘𝑦))) ∘ (𝑥(2nd ‘𝐹)𝑦)))
25		eqid 2738	. . . . . . . 8 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
26	12	3ad2ant1 1131	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → (1st ‘𝐹)(𝐶 Func 𝐷)(2nd ‘𝐹))
27		simp2 1135	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → 𝑥 ∈ (Base‘𝐶))
28		simp3 1136	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → 𝑦 ∈ (Base‘𝐶))
29	9, 25, 18, 26, 27, 28	funcf2 17499	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → (𝑥(2nd ‘𝐹)𝑦):(𝑥(Hom ‘𝐶)𝑦)⟶(((1st ‘𝐹)‘𝑥)(Hom ‘𝐷)((1st ‘𝐹)‘𝑦)))
30		fcoi2 6633	. . . . . . 7 ⊢ ((𝑥(2nd ‘𝐹)𝑦):(𝑥(Hom ‘𝐶)𝑦)⟶(((1st ‘𝐹)‘𝑥)(Hom ‘𝐷)((1st ‘𝐹)‘𝑦)) → (( I ↾ (((1st ‘𝐹)‘𝑥)(Hom ‘𝐷)((1st ‘𝐹)‘𝑦))) ∘ (𝑥(2nd ‘𝐹)𝑦)) = (𝑥(2nd ‘𝐹)𝑦))
31	29, 30	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → (( I ↾ (((1st ‘𝐹)‘𝑥)(Hom ‘𝐷)((1st ‘𝐹)‘𝑦))) ∘ (𝑥(2nd ‘𝐹)𝑦)) = (𝑥(2nd ‘𝐹)𝑦))
32	24, 31	eqtrd 2778	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → ((((1st ‘𝐹)‘𝑥)(2nd ‘𝐼)((1st ‘𝐹)‘𝑦)) ∘ (𝑥(2nd ‘𝐹)𝑦)) = (𝑥(2nd ‘𝐹)𝑦))
33	32	mpoeq3dva 7330	. . . 4 ⊢ (𝜑 → (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ ((((1st ‘𝐹)‘𝑥)(2nd ‘𝐼)((1st ‘𝐹)‘𝑦)) ∘ (𝑥(2nd ‘𝐹)𝑦))) = (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ (𝑥(2nd ‘𝐹)𝑦)))
34	9, 12	funcfn2 17500	. . . . 5 ⊢ (𝜑 → (2nd ‘𝐹) Fn ((Base‘𝐶) × (Base‘𝐶)))
35		fnov 7383	. . . . 5 ⊢ ((2nd ‘𝐹) Fn ((Base‘𝐶) × (Base‘𝐶)) ↔ (2nd ‘𝐹) = (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ (𝑥(2nd ‘𝐹)𝑦)))
36	34, 35	sylib 217	. . . 4 ⊢ (𝜑 → (2nd ‘𝐹) = (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ (𝑥(2nd ‘𝐹)𝑦)))
37	33, 36	eqtr4d 2781	. . 3 ⊢ (𝜑 → (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ ((((1st ‘𝐹)‘𝑥)(2nd ‘𝐼)((1st ‘𝐹)‘𝑦)) ∘ (𝑥(2nd ‘𝐹)𝑦))) = (2nd ‘𝐹))
38	16, 37	opeq12d 4809	. 2 ⊢ (𝜑 → ⟨((1st ‘𝐼) ∘ (1st ‘𝐹)), (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ ((((1st ‘𝐹)‘𝑥)(2nd ‘𝐼)((1st ‘𝐹)‘𝑦)) ∘ (𝑥(2nd ‘𝐹)𝑦)))⟩ = ⟨(1st ‘𝐹), (2nd ‘𝐹)⟩)
39	1	idfucl 17512	. . . 4 ⊢ (𝐷 ∈ Cat → 𝐼 ∈ (𝐷 Func 𝐷))
40	6, 39	syl 17	. . 3 ⊢ (𝜑 → 𝐼 ∈ (𝐷 Func 𝐷))
41	9, 3, 40	cofuval 17513	. 2 ⊢ (𝜑 → (𝐼 ∘func 𝐹) = ⟨((1st ‘𝐼) ∘ (1st ‘𝐹)), (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ ((((1st ‘𝐹)‘𝑥)(2nd ‘𝐼)((1st ‘𝐹)‘𝑦)) ∘ (𝑥(2nd ‘𝐹)𝑦)))⟩)
42		1st2nd 7853	. . 3 ⊢ ((Rel (𝐶 Func 𝐷) ∧ 𝐹 ∈ (𝐶 Func 𝐷)) → 𝐹 = ⟨(1st ‘𝐹), (2nd ‘𝐹)⟩)
43	10, 3, 42	sylancr 586	. 2 ⊢ (𝜑 → 𝐹 = ⟨(1st ‘𝐹), (2nd ‘𝐹)⟩)
44	38, 41, 43	3eqtr4d 2788	1 ⊢ (𝜑 → (𝐼 ∘func 𝐹) = 𝐹)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1085   = wceq 1539   ∈ wcel 2108  ⟨cop 4564   class class class wbr 5070   I cid 5479   × cxp 5578   ↾ cres 5582   ∘ ccom 5584  Rel wrel 5585   Fn wfn 6413  ⟶wf 6414  ‘cfv 6418  (class class class)co 7255   ∈ cmpo 7257  1^st c1st 7802  2^nd c2nd 7803  Basecbs 16840  Hom chom 16899  Catccat 17290   Func cfunc 17485  id_funccidfu 17486   ∘_func ccofu 17487

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rmo 3071  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-riota 7212  df-ov 7258  df-oprab 7259  df-mpo 7260  df-1st 7804  df-2nd 7805  df-map 8575  df-ixp 8644  df-cat 17294  df-cid 17295  df-func 17489  df-idfu 17490  df-cofu 17491

This theorem is referenced by:  catccatid  17737