Theorem cofulid 17803

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 2731	. . . . . 6 ⊢ (Base‘𝐷) = (Base‘𝐷)
3		cofulid.g	. . . . . . . 8 ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))
4		funcrcl 17776	. . . . . . . 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 17792	. . . . 5 ⊢ (𝜑 → (1st ‘𝐼) = ( I ↾ (Base‘𝐷)))
8	7	coeq1d 5806	. . . 4 ⊢ (𝜑 → ((1st ‘𝐼) ∘ (1st ‘𝐹)) = (( I ↾ (Base‘𝐷)) ∘ (1st ‘𝐹)))
9		eqid 2731	. . . . . 6 ⊢ (Base‘𝐶) = (Base‘𝐶)
10		relfunc 17775	. . . . . . 7 ⊢ Rel (𝐶 Func 𝐷)
11		1st2ndbr 7980	. . . . . . 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	9, 2, 12	funcf1 17779	. . . . 5 ⊢ (𝜑 → (1st ‘𝐹):(Base‘𝐶)⟶(Base‘𝐷))
14		fcoi2 6704	. . . . 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 2766	. . 3 ⊢ (𝜑 → ((1st ‘𝐼) ∘ (1st ‘𝐹)) = (1st ‘𝐹))
17	6	3ad2ant1 1133	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → 𝐷 ∈ Cat)
18		eqid 2731	. . . . . . . 8 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
19	13	ffvelcdmda 7023	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((1st ‘𝐹)‘𝑥) ∈ (Base‘𝐷))
20	19	3adant3 1132	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → ((1st ‘𝐹)‘𝑥) ∈ (Base‘𝐷))
21	13	ffvelcdmda 7023	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (Base‘𝐶)) → ((1st ‘𝐹)‘𝑦) ∈ (Base‘𝐷))
22	21	3adant2 1131	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → ((1st ‘𝐹)‘𝑦) ∈ (Base‘𝐷))
23	1, 2, 17, 18, 20, 22	idfu2nd 17790	. . . . . . 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 5806	. . . . . 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 2731	. . . . . . . 8 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
26	12	3ad2ant1 1133	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → (1st ‘𝐹)(𝐶 Func 𝐷)(2nd ‘𝐹))
27		simp2 1137	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → 𝑥 ∈ (Base‘𝐶))
28		simp3 1138	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → 𝑦 ∈ (Base‘𝐶))
29	9, 25, 18, 26, 27, 28	funcf2 17781	. . . . . . 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 6704	. . . . . . 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 2766	. . . . 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 7429	. . . 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 17782	. . . . 5 ⊢ (𝜑 → (2nd ‘𝐹) Fn ((Base‘𝐶) × (Base‘𝐶)))
35		fnov 7483	. . . . 5 ⊢ ((2nd ‘𝐹) Fn ((Base‘𝐶) × (Base‘𝐶)) ↔ (2nd ‘𝐹) = (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ (𝑥(2nd ‘𝐹)𝑦)))
36	34, 35	sylib 218	. . . 4 ⊢ (𝜑 → (2nd ‘𝐹) = (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ (𝑥(2nd ‘𝐹)𝑦)))
37	33, 36	eqtr4d 2769	. . 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 4832	. 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 17794	. . . 4 ⊢ (𝐷 ∈ Cat → 𝐼 ∈ (𝐷 Func 𝐷))
40	6, 39	syl 17	. . 3 ⊢ (𝜑 → 𝐼 ∈ (𝐷 Func 𝐷))
41	9, 3, 40	cofuval 17795	. 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 7977	. . 3 ⊢ ((Rel (𝐶 Func 𝐷) ∧ 𝐹 ∈ (𝐶 Func 𝐷)) → 𝐹 = ⟨(1st ‘𝐹), (2nd ‘𝐹)⟩)
43	10, 3, 42	sylancr 587	. 2 ⊢ (𝜑 → 𝐹 = ⟨(1st ‘𝐹), (2nd ‘𝐹)⟩)
44	38, 41, 43	3eqtr4d 2776	1 ⊢ (𝜑 → (𝐼 ∘func 𝐹) = 𝐹)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1541   ∈ wcel 2111  ⟨cop 4581   class class class wbr 5093   I cid 5513   × cxp 5617   ↾ cres 5621   ∘ ccom 5623  Rel wrel 5624   Fn wfn 6482  ⟶wf 6483  ‘cfv 6487  (class class class)co 7352   ∈ cmpo 7354  1^st c1st 7925  2^nd c2nd 7926  Basecbs 17126  Hom chom 17178  Catccat 17576   Func cfunc 17767  id_funccidfu 17768   ∘_func ccofu 17769

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5219  ax-sep 5236  ax-nul 5246  ax-pow 5305  ax-pr 5372  ax-un 7674

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rmo 3346  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4283  df-if 4475  df-pw 4551  df-sn 4576  df-pr 4578  df-op 4582  df-uni 4859  df-iun 4943  df-br 5094  df-opab 5156  df-mpt 5175  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 6443  df-fun 6489  df-fn 6490  df-f 6491  df-f1 6492  df-fo 6493  df-f1o 6494  df-fv 6495  df-riota 7309  df-ov 7355  df-oprab 7356  df-mpo 7357  df-1st 7927  df-2nd 7928  df-map 8758  df-ixp 8828  df-cat 17580  df-cid 17581  df-func 17771  df-idfu 17772  df-cofu 17773

This theorem is referenced by:  catccatid  18019  uobeqw  49325