Theorem cofurid 17798

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 2729	. . . . . 6 ⊢ (Base‘𝐶) = (Base‘𝐶)
3		cofulid.g	. . . . . . . 8 ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))
4		funcrcl 17770	. . . . . . . 8 ⊢ (𝐹 ∈ (𝐶 Func 𝐷) → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
5	3, 4	syl 17	. . . . . . 7 ⊢ (𝜑 → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
6	5	simpld 494	. . . . . 6 ⊢ (𝜑 → 𝐶 ∈ Cat)
7	1, 2, 6	idfu1st 17786	. . . . 5 ⊢ (𝜑 → (1st ‘𝐼) = ( I ↾ (Base‘𝐶)))
8	7	coeq2d 5805	. . . 4 ⊢ (𝜑 → ((1st ‘𝐹) ∘ (1st ‘𝐼)) = ((1st ‘𝐹) ∘ ( I ↾ (Base‘𝐶))))
9		eqid 2729	. . . . . 6 ⊢ (Base‘𝐷) = (Base‘𝐷)
10		relfunc 17769	. . . . . . 7 ⊢ Rel (𝐶 Func 𝐷)
11		1st2ndbr 7977	. . . . . . 7 ⊢ ((Rel (𝐶 Func 𝐷) ∧ 𝐹 ∈ (𝐶 Func 𝐷)) → (1st ‘𝐹)(𝐶 Func 𝐷)(2nd ‘𝐹))
12	10, 3, 11	sylancr 587	. . . . . 6 ⊢ (𝜑 → (1st ‘𝐹)(𝐶 Func 𝐷)(2nd ‘𝐹))
13	2, 9, 12	funcf1 17773	. . . . 5 ⊢ (𝜑 → (1st ‘𝐹):(Base‘𝐶)⟶(Base‘𝐷))
14		fcoi1 6698	. . . . 5 ⊢ ((1st ‘𝐹):(Base‘𝐶)⟶(Base‘𝐷) → ((1st ‘𝐹) ∘ ( I ↾ (Base‘𝐶))) = (1st ‘𝐹))
15	13, 14	syl 17	. . . 4 ⊢ (𝜑 → ((1st ‘𝐹) ∘ ( I ↾ (Base‘𝐶))) = (1st ‘𝐹))
16	8, 15	eqtrd 2764	. . 3 ⊢ (𝜑 → ((1st ‘𝐹) ∘ (1st ‘𝐼)) = (1st ‘𝐹))
17	7	3ad2ant1 1133	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → (1st ‘𝐼) = ( I ↾ (Base‘𝐶)))
18	17	fveq1d 6824	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → ((1st ‘𝐼)‘𝑥) = (( I ↾ (Base‘𝐶))‘𝑥))
19		fvresi 7109	. . . . . . . . . 10 ⊢ (𝑥 ∈ (Base‘𝐶) → (( I ↾ (Base‘𝐶))‘𝑥) = 𝑥)
20	19	3ad2ant2 1134	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → (( I ↾ (Base‘𝐶))‘𝑥) = 𝑥)
21	18, 20	eqtrd 2764	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → ((1st ‘𝐼)‘𝑥) = 𝑥)
22	17	fveq1d 6824	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → ((1st ‘𝐼)‘𝑦) = (( I ↾ (Base‘𝐶))‘𝑦))
23		fvresi 7109	. . . . . . . . . 10 ⊢ (𝑦 ∈ (Base‘𝐶) → (( I ↾ (Base‘𝐶))‘𝑦) = 𝑦)
24	23	3ad2ant3 1135	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → (( I ↾ (Base‘𝐶))‘𝑦) = 𝑦)
25	22, 24	eqtrd 2764	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → ((1st ‘𝐼)‘𝑦) = 𝑦)
26	21, 25	oveq12d 7367	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → (((1st ‘𝐼)‘𝑥)(2nd ‘𝐹)((1st ‘𝐼)‘𝑦)) = (𝑥(2nd ‘𝐹)𝑦))
27	6	3ad2ant1 1133	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → 𝐶 ∈ Cat)
28		eqid 2729	. . . . . . . 8 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
29		simp2 1137	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → 𝑥 ∈ (Base‘𝐶))
30		simp3 1138	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → 𝑦 ∈ (Base‘𝐶))
31	1, 2, 27, 28, 29, 30	idfu2nd 17784	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → (𝑥(2nd ‘𝐼)𝑦) = ( I ↾ (𝑥(Hom ‘𝐶)𝑦)))
32	26, 31	coeq12d 5807	. . . . . 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 2729	. . . . . . . 8 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
34	12	3ad2ant1 1133	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → (1st ‘𝐹)(𝐶 Func 𝐷)(2nd ‘𝐹))
35	2, 28, 33, 34, 29, 30	funcf2 17775	. . . . . . 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 6698	. . . . . . 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 17	. . . . . 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 2764	. . . . 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 7426	. . . 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 17776	. . . . 5 ⊢ (𝜑 → (2nd ‘𝐹) Fn ((Base‘𝐶) × (Base‘𝐶)))
41		fnov 7480	. . . . 5 ⊢ ((2nd ‘𝐹) Fn ((Base‘𝐶) × (Base‘𝐶)) ↔ (2nd ‘𝐹) = (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ (𝑥(2nd ‘𝐹)𝑦)))
42	40, 41	sylib 218	. . . 4 ⊢ (𝜑 → (2nd ‘𝐹) = (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ (𝑥(2nd ‘𝐹)𝑦)))
43	39, 42	eqtr4d 2767	. . 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 4832	. 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 17788	. . . 4 ⊢ (𝐶 ∈ Cat → 𝐼 ∈ (𝐶 Func 𝐶))
46	6, 45	syl 17	. . 3 ⊢ (𝜑 → 𝐼 ∈ (𝐶 Func 𝐶))
47	2, 46, 3	cofuval 17789	. 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 7974	. . 3 ⊢ ((Rel (𝐶 Func 𝐷) ∧ 𝐹 ∈ (𝐶 Func 𝐷)) → 𝐹 = ⟨(1st ‘𝐹), (2nd ‘𝐹)⟩)
49	10, 3, 48	sylancr 587	. 2 ⊢ (𝜑 → 𝐹 = ⟨(1st ‘𝐹), (2nd ‘𝐹)⟩)
50	44, 47, 49	3eqtr4d 2774	1 ⊢ (𝜑 → (𝐹 ∘func 𝐼) = 𝐹)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  ⟨cop 4583   class class class wbr 5092   I cid 5513   × cxp 5617   ↾ cres 5621   ∘ ccom 5623  Rel wrel 5624   Fn wfn 6477  ⟶wf 6478  ‘cfv 6482  (class class class)co 7349   ∈ cmpo 7351  1^st c1st 7922  2^nd c2nd 7923  Basecbs 17120  Hom chom 17172  Catccat 17570   Func cfunc 17761  id_funccidfu 17762   ∘_func ccofu 17763

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5218  ax-sep 5235  ax-nul 5245  ax-pow 5304  ax-pr 5371  ax-un 7671

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rmo 3343  df-reu 3344  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5174  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-f1 6487  df-fo 6488  df-f1o 6489  df-fv 6490  df-riota 7306  df-ov 7352  df-oprab 7353  df-mpo 7354  df-1st 7924  df-2nd 7925  df-map 8755  df-ixp 8825  df-cat 17574  df-cid 17575  df-func 17765  df-idfu 17766  df-cofu 17767

This theorem is referenced by:  catccatid  18013