Theorem arwlid 18034

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 2737	. . . . . 6 ⊢ (Base‘𝐶) = (Base‘𝐶)
3		arwlid.f	. . . . . . 7 ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐻𝑌))
4		arwlid.h	. . . . . . . 8 ⊢ 𝐻 = (Homa‘𝐶)
5	4	homarcl 17990	. . . . . . 7 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → 𝐶 ∈ Cat)
6	3, 5	syl 17	. . . . . 6 ⊢ (𝜑 → 𝐶 ∈ Cat)
7		eqid 2737	. . . . . 6 ⊢ (Id‘𝐶) = (Id‘𝐶)
8	4, 2	homarcl2 17997	. . . . . . . 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 495	. . . . . 6 ⊢ (𝜑 → 𝑌 ∈ (Base‘𝐶))
11	1, 2, 6, 7, 10	ida2 18021	. . . . 5 ⊢ (𝜑 → (2nd ‘( 1 ‘𝑌)) = ((Id‘𝐶)‘𝑌))
12	11	oveq1d 7377	. . . 4 ⊢ (𝜑 → ((2nd ‘( 1 ‘𝑌))(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑌)(2nd ‘𝐹)) = (((Id‘𝐶)‘𝑌)(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑌)(2nd ‘𝐹)))
13		eqid 2737	. . . . 5 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
14	9	simpld 494	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ (Base‘𝐶))
15		eqid 2737	. . . . 5 ⊢ (comp‘𝐶) = (comp‘𝐶)
16	4, 13	homahom 18001	. . . . . 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 17644	. . . 4 ⊢ (𝜑 → (((Id‘𝐶)‘𝑌)(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑌)(2nd ‘𝐹)) = (2nd ‘𝐹))
19	12, 18	eqtrd 2772	. . 3 ⊢ (𝜑 → ((2nd ‘( 1 ‘𝑌))(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑌)(2nd ‘𝐹)) = (2nd ‘𝐹))
20	19	oteq3d 4831	. 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 18022	. . 3 ⊢ (𝜑 → ( 1 ‘𝑌) ∈ (𝑌𝐻𝑌))
23	21, 4, 3, 22, 15	coaval 18030	. 2 ⊢ (𝜑 → (( 1 ‘𝑌) · 𝐹) = ⟨𝑋, 𝑌, ((2nd ‘( 1 ‘𝑌))(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑌)(2nd ‘𝐹))⟩)
24	4	homadmcd 18004	. . 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 2782	1 ⊢ (𝜑 → (( 1 ‘𝑌) · 𝐹) = 𝐹)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ⟨cop 4574  ⟨cotp 4576  ‘cfv 6494  (class class class)co 7362  2^nd c2nd 7936  Basecbs 17174  Hom chom 17226  compcco 17227  Catccat 17625  Idccid 17626  Hom_achoma 17985  Id_acida 18015  comp_accoa 18016

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5213  ax-sep 5232  ax-nul 5242  ax-pow 5304  ax-pr 5372  ax-un 7684

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rmo 3343  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-ot 4577  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5521  df-xp 5632  df-rel 5633  df-cnv 5634  df-co 5635  df-dm 5636  df-rn 5637  df-res 5638  df-ima 5639  df-iota 6450  df-fun 6496  df-fn 6497  df-f 6498  df-f1 6499  df-fo 6500  df-f1o 6501  df-fv 6502  df-riota 7319  df-ov 7365  df-oprab 7366  df-mpo 7367  df-1st 7937  df-2nd 7938  df-cat 17629  df-cid 17630  df-doma 17986  df-coda 17987  df-homa 17988  df-arw 17989  df-ida 18017  df-coa 18018

This theorem is referenced by: (None)