Theorem cofulid 16550

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 2622	. . . . . 6 ⊢ (Base‘𝐷) = (Base‘𝐷)
3		cofulid.g	. . . . . . . 8 ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))
4		funcrcl 16523	. . . . . . . 8 ⊢ (𝐹 ∈ (𝐶 Func 𝐷) → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
5	3, 4	syl 17	. . . . . . 7 ⊢ (𝜑 → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
6	5	simprd 479	. . . . . 6 ⊢ (𝜑 → 𝐷 ∈ Cat)
7	1, 2, 6	idfu1st 16539	. . . . 5 ⊢ (𝜑 → (1st ‘𝐼) = ( I ↾ (Base‘𝐷)))
8	7	coeq1d 5283	. . . 4 ⊢ (𝜑 → ((1st ‘𝐼) ∘ (1st ‘𝐹)) = (( I ↾ (Base‘𝐷)) ∘ (1st ‘𝐹)))
9		eqid 2622	. . . . . 6 ⊢ (Base‘𝐶) = (Base‘𝐶)
10		relfunc 16522	. . . . . . 7 ⊢ Rel (𝐶 Func 𝐷)
11		1st2ndbr 7217	. . . . . . 7 ⊢ ((Rel (𝐶 Func 𝐷) ∧ 𝐹 ∈ (𝐶 Func 𝐷)) → (1st ‘𝐹)(𝐶 Func 𝐷)(2nd ‘𝐹))
12	10, 3, 11	sylancr 695	. . . . . 6 ⊢ (𝜑 → (1st ‘𝐹)(𝐶 Func 𝐷)(2nd ‘𝐹))
13	9, 2, 12	funcf1 16526	. . . . 5 ⊢ (𝜑 → (1st ‘𝐹):(Base‘𝐶)⟶(Base‘𝐷))
14		fcoi2 6079	. . . . 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 2656	. . 3 ⊢ (𝜑 → ((1st ‘𝐼) ∘ (1st ‘𝐹)) = (1st ‘𝐹))
17	6	3ad2ant1 1082	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → 𝐷 ∈ Cat)
18		eqid 2622	. . . . . . . 8 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
19	13	ffvelrnda 6359	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((1st ‘𝐹)‘𝑥) ∈ (Base‘𝐷))
20	19	3adant3 1081	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → ((1st ‘𝐹)‘𝑥) ∈ (Base‘𝐷))
21	13	ffvelrnda 6359	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (Base‘𝐶)) → ((1st ‘𝐹)‘𝑦) ∈ (Base‘𝐷))
22	21	3adant2 1080	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → ((1st ‘𝐹)‘𝑦) ∈ (Base‘𝐷))
23	1, 2, 17, 18, 20, 22	idfu2nd 16537	. . . . . . 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 5283	. . . . . 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 2622	. . . . . . . 8 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
26	12	3ad2ant1 1082	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → (1st ‘𝐹)(𝐶 Func 𝐷)(2nd ‘𝐹))
27		simp2 1062	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → 𝑥 ∈ (Base‘𝐶))
28		simp3 1063	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → 𝑦 ∈ (Base‘𝐶))
29	9, 25, 18, 26, 27, 28	funcf2 16528	. . . . . . 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 6079	. . . . . . 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 2656	. . . . 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	mpt2eq3dva 6719	. . . 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 16529	. . . . 5 ⊢ (𝜑 → (2nd ‘𝐹) Fn ((Base‘𝐶) × (Base‘𝐶)))
35		fnov 6768	. . . . 5 ⊢ ((2nd ‘𝐹) Fn ((Base‘𝐶) × (Base‘𝐶)) ↔ (2nd ‘𝐹) = (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ (𝑥(2nd ‘𝐹)𝑦)))
36	34, 35	sylib 208	. . . 4 ⊢ (𝜑 → (2nd ‘𝐹) = (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ (𝑥(2nd ‘𝐹)𝑦)))
37	33, 36	eqtr4d 2659	. . 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 4410	. 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 16541	. . . 4 ⊢ (𝐷 ∈ Cat → 𝐼 ∈ (𝐷 Func 𝐷))
40	6, 39	syl 17	. . 3 ⊢ (𝜑 → 𝐼 ∈ (𝐷 Func 𝐷))
41	9, 3, 40	cofuval 16542	. 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 7214	. . 3 ⊢ ((Rel (𝐶 Func 𝐷) ∧ 𝐹 ∈ (𝐶 Func 𝐷)) → 𝐹 = ⟨(1st ‘𝐹), (2nd ‘𝐹)⟩)
43	10, 3, 42	sylancr 695	. 2 ⊢ (𝜑 → 𝐹 = ⟨(1st ‘𝐹), (2nd ‘𝐹)⟩)
44	38, 41, 43	3eqtr4d 2666	1 ⊢ (𝜑 → (𝐼 ∘func 𝐹) = 𝐹)

Syntax hints:   → wi 4   ∧ wa 384   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990  ⟨cop 4183   class class class wbr 4653   I cid 5023   × cxp 5112   ↾ cres 5116   ∘ ccom 5118  Rel wrel 5119   Fn wfn 5883  ⟶wf 5884  ‘cfv 5888  (class class class)co 6650   ↦ cmpt2 6652  1^st c1st 7166  2^nd c2nd 7167  Basecbs 15857  Hom chom 15952  Catccat 16325   Func cfunc 16514  id_funccidfu 16515   ∘_func ccofu 16516

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-map 7859  df-ixp 7909  df-cat 16329  df-cid 16330  df-func 16518  df-idfu 16519  df-cofu 16520

This theorem is referenced by:  catccatid  16752