Theorem arwlid 18107

Description: Left identity of a category using arrow notation. (Contributed by Mario Carneiro, 11-Jan-2017.)

Hypotheses

Ref	Expression
arwlid.h	⊢ 𝐻 = (Homa‘𝐶)
arwlid.o	⊢ · = (compa‘𝐶)
arwlid.a	⊢ 1 = (Ida‘𝐶)
arwlid.f	⊢ (𝜑 → 𝐹 ∈ (𝑋𝐻𝑌))

Assertion

Ref	Expression
arwlid	⊢ (𝜑 → (( 1 ‘𝑌) · 𝐹) = 𝐹)

Proof of Theorem arwlid

Step	Hyp	Ref	Expression
1		arwlid.a	. . . . . 6 ⊢ 1 = (Ida‘𝐶)
2		eqid 2764	. . . . . 6 ⊢ (Base‘𝐶) = (Base‘𝐶)
3		arwlid.f	. . . . . . 7 ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐻𝑌))
4		arwlid.h	. . . . . . . 8 ⊢ 𝐻 = (Homa‘𝐶)
5	4	homarcl 18063	. . . . . . 7 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → 𝐶 ∈ Cat)
6	3, 5	syl 17	. . . . . 6 ⊢ (𝜑 → 𝐶 ∈ Cat)
7		eqid 2764	. . . . . 6 ⊢ (Id‘𝐶) = (Id‘𝐶)
8	4, 2	homarcl2 18070	. . . . . . . 8 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → (𝑋 ∈ (Base‘𝐶) ∧ 𝑌 ∈ (Base‘𝐶)))
9	3, 8	syl 17	. . . . . . 7 ⊢ (𝜑 → (𝑋 ∈ (Base‘𝐶) ∧ 𝑌 ∈ (Base‘𝐶)))
10	9	simprd 499	. . . . . 6 ⊢ (𝜑 → 𝑌 ∈ (Base‘𝐶))
11	1, 2, 6, 7, 10	ida2 18094	. . . . 5 ⊢ (𝜑 → (2nd ‘( 1 ‘𝑌)) = ((Id‘𝐶)‘𝑌))
12	11	oveq1d 7413	. . . 4 ⊢ (𝜑 → ((2nd ‘( 1 ‘𝑌))(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑌)(2nd ‘𝐹)) = (((Id‘𝐶)‘𝑌)(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑌)(2nd ‘𝐹)))
13		eqid 2764	. . . . 5 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
14	9	simpld 498	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ (Base‘𝐶))
15		eqid 2764	. . . . 5 ⊢ (comp‘𝐶) = (comp‘𝐶)
16	4, 13	homahom 18074	. . . . . 6 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → (2nd ‘𝐹) ∈ (𝑋(Hom ‘𝐶)𝑌))
17	3, 16	syl 17	. . . . 5 ⊢ (𝜑 → (2nd ‘𝐹) ∈ (𝑋(Hom ‘𝐶)𝑌))
18	2, 13, 7, 6, 14, 15, 10, 17	catlid 17717	. . . 4 ⊢ (𝜑 → (((Id‘𝐶)‘𝑌)(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑌)(2nd ‘𝐹)) = (2nd ‘𝐹))
19	12, 18	eqtrd 2799	. . 3 ⊢ (𝜑 → ((2nd ‘( 1 ‘𝑌))(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑌)(2nd ‘𝐹)) = (2nd ‘𝐹))
20	19	oteq3d 4847	. 2 ⊢ (𝜑 → ⟨𝑋, 𝑌, ((2nd ‘( 1 ‘𝑌))(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑌)(2nd ‘𝐹))⟩ = ⟨𝑋, 𝑌, (2nd ‘𝐹)⟩)
21		arwlid.o	. . 3 ⊢ · = (compa‘𝐶)
22	1, 2, 6, 10, 4	idahom 18095	. . 3 ⊢ (𝜑 → ( 1 ‘𝑌) ∈ (𝑌𝐻𝑌))
23	21, 4, 3, 22, 15	coaval 18103	. 2 ⊢ (𝜑 → (( 1 ‘𝑌) · 𝐹) = ⟨𝑋, 𝑌, ((2nd ‘( 1 ‘𝑌))(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑌)(2nd ‘𝐹))⟩)
24	4	homadmcd 18077	. . 3 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → 𝐹 = ⟨𝑋, 𝑌, (2nd ‘𝐹)⟩)
25	3, 24	syl 17	. 2 ⊢ (𝜑 → 𝐹 = ⟨𝑋, 𝑌, (2nd ‘𝐹)⟩)
26	20, 23, 25	3eqtr4d 2809	1 ⊢ (𝜑 → (( 1 ‘𝑌) · 𝐹) = 𝐹)

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1562   ∈ wcel 2144  ⟨cop 4590  ⟨cotp 4592  ‘cfv 6523  (class class class)co 7398  2^nd c2nd 7971  Basecbs 17247  Hom chom 17299  compcco 17300  Catccat 17698  Idccid 17699  Hom_achoma 18058  Id_acida 18088  comp_accoa 18089

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2214  ax-ext 2736  ax-rep 5229  ax-sep 5248  ax-nul 5258  ax-pow 5324  ax-pr 5392  ax-un 7720

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1101  df-tru 1565  df-fal 1575  df-ex 1802  df-nf 1806  df-sb 2093  df-mo 2568  df-eu 2598  df-clab 2743  df-cleq 2756  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3079  df-rex 3089  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3458  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-ot 4593  df-uni 4868  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-id 5544  df-xp 5655  df-rel 5656  df-cnv 5657  df-co 5658  df-dm 5659  df-rn 5660  df-res 5661  df-ima 5662  df-iota 6479  df-fun 6525  df-fn 6526  df-f 6527  df-f1 6528  df-fo 6529  df-f1o 6530  df-fv 6531  df-riota 7355  df-ov 7401  df-oprab 7402  df-mpo 7403  df-1st 7972  df-2nd 7973  df-cat 17702  df-cid 17703  df-doma 18059  df-coda 18060  df-homa 18061  df-arw 18062  df-ida 18090  df-coa 18091

This theorem is referenced by: (None)