Theorem cofurid 18081

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

Hypotheses

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

Assertion

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

Proof of Theorem cofurid

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

Step	Hyp	Ref	Expression
1		cofurid.1	. . . . . 6 ⊢ 𝐼 = (idfunc‘𝐶)
2		eqid 2763	. . . . . 6 ⊢ (Base‘𝐶) = (Base‘𝐶)
3		cofulid.g	. . . . . . . 8 ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))
4		funcrcl 18053	. . . . . . . 8 ⊢ (𝐹 ∈ (𝐶 Func 𝐷) → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
5	3, 4	syl 18	. . . . . . 7 ⊢ (𝜑 → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
6	5	simpld 500	. . . . . 6 ⊢ (𝜑 → 𝐶 ∈ Cat)
7	1, 2, 6	idfu1st 18069	. . . . 5 ⊢ (𝜑 → (1st ‘𝐼) = ( I ↾ (Base‘𝐶)))
8	7	coeq2d 5845	. . . 4 ⊢ (𝜑 → ((1st ‘𝐹) ∘ (1st ‘𝐼)) = ((1st ‘𝐹) ∘ ( I ↾ (Base‘𝐶))))
9		eqid 2763	. . . . . 6 ⊢ (Base‘𝐷) = (Base‘𝐷)
10		relfunc 18052	. . . . . . 7 ⊢ Rel (𝐶 Func 𝐷)
11		1st2ndbr 8063	. . . . . . 7 ⊢ ((Rel (𝐶 Func 𝐷) ∧ 𝐹 ∈ (𝐶 Func 𝐷)) → (1st ‘𝐹)(𝐶 Func 𝐷)(2nd ‘𝐹))
12	10, 3, 11	sylancr 599	. . . . . 6 ⊢ (𝜑 → (1st ‘𝐹)(𝐶 Func 𝐷)(2nd ‘𝐹))
13	2, 9, 12	funcf1 18056	. . . . 5 ⊢ (𝜑 → (1st ‘𝐹):(Base‘𝐶)⟶(Base‘𝐷))
14		fcoi1 6767	. . . . 5 ⊢ ((1st ‘𝐹):(Base‘𝐶)⟶(Base‘𝐷) → ((1st ‘𝐹) ∘ ( I ↾ (Base‘𝐶))) = (1st ‘𝐹))
15	13, 14	syl 18	. . . 4 ⊢ (𝜑 → ((1st ‘𝐹) ∘ ( I ↾ (Base‘𝐶))) = (1st ‘𝐹))
16	8, 15	eqtrd 2798	. . 3 ⊢ (𝜑 → ((1st ‘𝐹) ∘ (1st ‘𝐼)) = (1st ‘𝐹))
17	7	3ad2ant1 1151	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → (1st ‘𝐼) = ( I ↾ (Base‘𝐶)))
18	17	fveq1d 6898	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → ((1st ‘𝐼)‘𝑥) = (( I ↾ (Base‘𝐶))‘𝑥))
19		fvresi 7188	. . . . . . . . . 10 ⊢ (𝑥 ∈ (Base‘𝐶) → (( I ↾ (Base‘𝐶))‘𝑥) = 𝑥)
20	19	3ad2ant2 1152	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → (( I ↾ (Base‘𝐶))‘𝑥) = 𝑥)
21	18, 20	eqtrd 2798	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → ((1st ‘𝐼)‘𝑥) = 𝑥)
22	17	fveq1d 6898	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → ((1st ‘𝐼)‘𝑦) = (( I ↾ (Base‘𝐶))‘𝑦))
23		fvresi 7188	. . . . . . . . . 10 ⊢ (𝑦 ∈ (Base‘𝐶) → (( I ↾ (Base‘𝐶))‘𝑦) = 𝑦)
24	23	3ad2ant3 1153	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → (( I ↾ (Base‘𝐶))‘𝑦) = 𝑦)
25	22, 24	eqtrd 2798	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → ((1st ‘𝐼)‘𝑦) = 𝑦)
26	21, 25	oveq12d 7448	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → (((1st ‘𝐼)‘𝑥)(2nd ‘𝐹)((1st ‘𝐼)‘𝑦)) = (𝑥(2nd ‘𝐹)𝑦))
27	6	3ad2ant1 1151	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → 𝐶 ∈ Cat)
28		eqid 2763	. . . . . . . 8 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
29		simp2 1155	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → 𝑥 ∈ (Base‘𝐶))
30		simp3 1156	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → 𝑦 ∈ (Base‘𝐶))
31	1, 2, 27, 28, 29, 30	idfu2nd 18067	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → (𝑥(2nd ‘𝐼)𝑦) = ( I ↾ (𝑥(Hom ‘𝐶)𝑦)))
32	26, 31	coeq12d 5847	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → ((((1st ‘𝐼)‘𝑥)(2nd ‘𝐹)((1st ‘𝐼)‘𝑦)) ∘ (𝑥(2nd ‘𝐼)𝑦)) = ((𝑥(2nd ‘𝐹)𝑦) ∘ ( I ↾ (𝑥(Hom ‘𝐶)𝑦))))
33		eqid 2763	. . . . . . . 8 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
34	12	3ad2ant1 1151	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → (1st ‘𝐹)(𝐶 Func 𝐷)(2nd ‘𝐹))
35	2, 28, 33, 34, 29, 30	funcf2 18058	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → (𝑥(2nd ‘𝐹)𝑦):(𝑥(Hom ‘𝐶)𝑦)⟶(((1st ‘𝐹)‘𝑥)(Hom ‘𝐷)((1st ‘𝐹)‘𝑦)))
36		fcoi1 6767	. . . . . . 7 ⊢ ((𝑥(2nd ‘𝐹)𝑦):(𝑥(Hom ‘𝐶)𝑦)⟶(((1st ‘𝐹)‘𝑥)(Hom ‘𝐷)((1st ‘𝐹)‘𝑦)) → ((𝑥(2nd ‘𝐹)𝑦) ∘ ( I ↾ (𝑥(Hom ‘𝐶)𝑦))) = (𝑥(2nd ‘𝐹)𝑦))
37	35, 36	syl 18	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → ((𝑥(2nd ‘𝐹)𝑦) ∘ ( I ↾ (𝑥(Hom ‘𝐶)𝑦))) = (𝑥(2nd ‘𝐹)𝑦))
38	32, 37	eqtrd 2798	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → ((((1st ‘𝐼)‘𝑥)(2nd ‘𝐹)((1st ‘𝐼)‘𝑦)) ∘ (𝑥(2nd ‘𝐼)𝑦)) = (𝑥(2nd ‘𝐹)𝑦))
39	38	mpoeq3dva 7507	. . . 4 ⊢ (𝜑 → (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ ((((1st ‘𝐼)‘𝑥)(2nd ‘𝐹)((1st ‘𝐼)‘𝑦)) ∘ (𝑥(2nd ‘𝐼)𝑦))) = (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ (𝑥(2nd ‘𝐹)𝑦)))
40	2, 12	funcfn2 18059	. . . . 5 ⊢ (𝜑 → (2nd ‘𝐹) Fn ((Base‘𝐶) × (Base‘𝐶)))
41		fnov 7561	. . . . 5 ⊢ ((2nd ‘𝐹) Fn ((Base‘𝐶) × (Base‘𝐶)) ↔ (2nd ‘𝐹) = (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ (𝑥(2nd ‘𝐹)𝑦)))
42	40, 41	sylib 221	. . . 4 ⊢ (𝜑 → (2nd ‘𝐹) = (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ (𝑥(2nd ‘𝐹)𝑦)))
43	39, 42	eqtr4d 2801	. . 3 ⊢ (𝜑 → (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ ((((1st ‘𝐼)‘𝑥)(2nd ‘𝐹)((1st ‘𝐼)‘𝑦)) ∘ (𝑥(2nd ‘𝐼)𝑦))) = (2nd ‘𝐹))
44	16, 43	opeq12d 4851	. 2 ⊢ (𝜑 → ⟨((1st ‘𝐹) ∘ (1st ‘𝐼)), (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ ((((1st ‘𝐼)‘𝑥)(2nd ‘𝐹)((1st ‘𝐼)‘𝑦)) ∘ (𝑥(2nd ‘𝐼)𝑦)))⟩ = ⟨(1st ‘𝐹), (2nd ‘𝐹)⟩)
45	1	idfucl 18071	. . . 4 ⊢ (𝐶 ∈ Cat → 𝐼 ∈ (𝐶 Func 𝐶))
46	6, 45	syl 18	. . 3 ⊢ (𝜑 → 𝐼 ∈ (𝐶 Func 𝐶))
47	2, 46, 3	cofuval 18072	. 2 ⊢ (𝜑 → (𝐹 ∘func 𝐼) = ⟨((1st ‘𝐹) ∘ (1st ‘𝐼)), (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ ((((1st ‘𝐼)‘𝑥)(2nd ‘𝐹)((1st ‘𝐼)‘𝑦)) ∘ (𝑥(2nd ‘𝐼)𝑦)))⟩)
48		1st2nd 8060	. . 3 ⊢ ((Rel (𝐶 Func 𝐷) ∧ 𝐹 ∈ (𝐶 Func 𝐷)) → 𝐹 = ⟨(1st ‘𝐹), (2nd ‘𝐹)⟩)
49	10, 3, 48	sylancr 599	. 2 ⊢ (𝜑 → 𝐹 = ⟨(1st ‘𝐹), (2nd ‘𝐹)⟩)
50	44, 47, 49	3eqtr4d 2808	1 ⊢ (𝜑 → (𝐹 ∘func 𝐼) = 𝐹)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1569   ∈ wcel 2146  ⟨cop 4838   class class class wbr 5106   I cid 5549   × cxp 5653   ↾ cres 5657   ∘ ccom 5659  Rel wrel 5660   Fn wfn 6542  ⟶wf 6543  ‘cfv 6547  (class class class)co 7430   ∈ cmpo 7432  1^st c1st 8009  2^nd c2nd 8010  Basecbs 17402  Hom chom 17454  Catccat 17853   Func cfunc 18044  id_funccidfu 18045   ∘_func ccofu 18046

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2196  ax-12 2215  ax-ext 2735  ax-rep 5235  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7761

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2101  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-iota 6501  df-fun 6549  df-fn 6550  df-f 6551  df-f1 6552  df-fo 6553  df-f1o 6554  df-fv 6555  df-riota 7387  df-ov 7433  df-oprab 7434  df-mpo 7435  df-1st 8011  df-2nd 8012  df-map 8863  df-ixp 8940  df-cat 17857  df-cid 17858  df-func 18048  df-idfu 18049  df-cofu 18050

This theorem is used by:  catccatid  18296  fucoaltvallem  50637