Theorem cofurid 17925

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 17897	. . . . . . . 8 ⊢ (𝐹 ∈ (𝐶 Func 𝐷) → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
5	3, 4	syl 17	. . . . . . 7 ⊢ (𝜑 → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
6	5	simpld 498	. . . . . 6 ⊢ (𝜑 → 𝐶 ∈ Cat)
7	1, 2, 6	idfu1st 17913	. . . . 5 ⊢ (𝜑 → (1st ‘𝐼) = ( I ↾ (Base‘𝐶)))
8	7	coeq2d 5835	. . . 4 ⊢ (𝜑 → ((1st ‘𝐹) ∘ (1st ‘𝐼)) = ((1st ‘𝐹) ∘ ( I ↾ (Base‘𝐶))))
9		eqid 2763	. . . . . 6 ⊢ (Base‘𝐷) = (Base‘𝐷)
10		relfunc 17896	. . . . . . 7 ⊢ Rel (𝐶 Func 𝐷)
11		1st2ndbr 8024	. . . . . . 7 ⊢ ((Rel (𝐶 Func 𝐷) ∧ 𝐹 ∈ (𝐶 Func 𝐷)) → (1st ‘𝐹)(𝐶 Func 𝐷)(2nd ‘𝐹))
12	10, 3, 11	sylancr 596	. . . . . 6 ⊢ (𝜑 → (1st ‘𝐹)(𝐶 Func 𝐷)(2nd ‘𝐹))
13	2, 9, 12	funcf1 17900	. . . . 5 ⊢ (𝜑 → (1st ‘𝐹):(Base‘𝐶)⟶(Base‘𝐷))
14		fcoi1 6739	. . . . 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 2798	. . 3 ⊢ (𝜑 → ((1st ‘𝐹) ∘ (1st ‘𝐼)) = (1st ‘𝐹))
17	7	3ad2ant1 1147	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → (1st ‘𝐼) = ( I ↾ (Base‘𝐶)))
18	17	fveq1d 6870	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → ((1st ‘𝐼)‘𝑥) = (( I ↾ (Base‘𝐶))‘𝑥))
19		fvresi 7158	. . . . . . . . . 10 ⊢ (𝑥 ∈ (Base‘𝐶) → (( I ↾ (Base‘𝐶))‘𝑥) = 𝑥)
20	19	3ad2ant2 1148	. . . . . . . . 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 6870	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → ((1st ‘𝐼)‘𝑦) = (( I ↾ (Base‘𝐶))‘𝑦))
23		fvresi 7158	. . . . . . . . . 10 ⊢ (𝑦 ∈ (Base‘𝐶) → (( I ↾ (Base‘𝐶))‘𝑦) = 𝑦)
24	23	3ad2ant3 1149	. . . . . . . . 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 7415	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → (((1st ‘𝐼)‘𝑥)(2nd ‘𝐹)((1st ‘𝐼)‘𝑦)) = (𝑥(2nd ‘𝐹)𝑦))
27	6	3ad2ant1 1147	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → 𝐶 ∈ Cat)
28		eqid 2763	. . . . . . . 8 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
29		simp2 1151	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → 𝑥 ∈ (Base‘𝐶))
30		simp3 1152	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → 𝑦 ∈ (Base‘𝐶))
31	1, 2, 27, 28, 29, 30	idfu2nd 17911	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → (𝑥(2nd ‘𝐼)𝑦) = ( I ↾ (𝑥(Hom ‘𝐶)𝑦)))
32	26, 31	coeq12d 5837	. . . . . 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 1147	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → (1st ‘𝐹)(𝐶 Func 𝐷)(2nd ‘𝐹))
35	2, 28, 33, 34, 29, 30	funcf2 17902	. . . . . . 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 6739	. . . . . . 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 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 7474	. . . 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 17903	. . . . 5 ⊢ (𝜑 → (2nd ‘𝐹) Fn ((Base‘𝐶) × (Base‘𝐶)))
41		fnov 7528	. . . . 5 ⊢ ((2nd ‘𝐹) Fn ((Base‘𝐶) × (Base‘𝐶)) ↔ (2nd ‘𝐹) = (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ (𝑥(2nd ‘𝐹)𝑦)))
42	40, 41	sylib 220	. . . 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 4840	. 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 17915	. . . 4 ⊢ (𝐶 ∈ Cat → 𝐼 ∈ (𝐶 Func 𝐶))
46	6, 45	syl 17	. . 3 ⊢ (𝜑 → 𝐼 ∈ (𝐶 Func 𝐶))
47	2, 46, 3	cofuval 17916	. 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 8021	. . 3 ⊢ ((Rel (𝐶 Func 𝐷) ∧ 𝐹 ∈ (𝐶 Func 𝐷)) → 𝐹 = ⟨(1st ‘𝐹), (2nd ‘𝐹)⟩)
49	10, 3, 48	sylancr 596	. 2 ⊢ (𝜑 → 𝐹 = ⟨(1st ‘𝐹), (2nd ‘𝐹)⟩)
50	44, 47, 49	3eqtr4d 2808	1 ⊢ (𝜑 → (𝐹 ∘func 𝐼) = 𝐹)

Syntax hints:   → wi 4   ∧ wa 399   ∧ w3a 1099   = wceq 1561   ∈ wcel 2143  ⟨cop 4589   class class class wbr 5101   I cid 5542   × cxp 5646   ↾ cres 5650   ∘ ccom 5652  Rel wrel 5653   Fn wfn 6517  ⟶wf 6518  ‘cfv 6522  (class class class)co 7397   ∈ cmpo 7399  1^st c1st 7969  2^nd c2nd 7970  Basecbs 17246  Hom chom 17298  Catccat 17697   Func cfunc 17888  id_funccidfu 17889   ∘_func ccofu 17890

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1816  ax-4 1830  ax-5 1931  ax-6 1988  ax-7 2029  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5228  ax-sep 5247  ax-nul 5257  ax-pow 5323  ax-pr 5391  ax-un 7719

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1101  df-tru 1564  df-fal 1574  df-ex 1801  df-nf 1805  df-sb 2092  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3078  df-rex 3088  df-rmo 3368  df-reu 3369  df-rab 3416  df-v 3457  df-sbc 3746  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-iun 4952  df-br 5102  df-opab 5164  df-mpt 5183  df-id 5543  df-xp 5654  df-rel 5655  df-cnv 5656  df-co 5657  df-dm 5658  df-rn 5659  df-res 5660  df-ima 5661  df-iota 6478  df-fun 6524  df-fn 6525  df-f 6526  df-f1 6527  df-fo 6528  df-f1o 6529  df-fv 6530  df-riota 7354  df-ov 7400  df-oprab 7401  df-mpo 7402  df-1st 7971  df-2nd 7972  df-map 8811  df-ixp 8881  df-cat 17701  df-cid 17702  df-func 17892  df-idfu 17893  df-cofu 17894

This theorem is referenced by:  catccatid  18140