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Theorem arwrid 18095
Description: Right identity of a category using arrow notation. (Contributed by Mario Carneiro, 11-Jan-2017.)
Hypotheses
Ref Expression
arwlid.h 𝐻 = (Homa𝐶)
arwlid.o · = (compa𝐶)
arwlid.a 1 = (Ida𝐶)
arwlid.f (𝜑𝐹 ∈ (𝑋𝐻𝑌))
Assertion
Ref Expression
arwrid (𝜑 → (𝐹 · ( 1𝑋)) = 𝐹)

Proof of Theorem arwrid
StepHypRef Expression
1 arwlid.a . . . . . 6 1 = (Ida𝐶)
2 eqid 2726 . . . . . 6 (Base‘𝐶) = (Base‘𝐶)
3 arwlid.f . . . . . . 7 (𝜑𝐹 ∈ (𝑋𝐻𝑌))
4 arwlid.h . . . . . . . 8 𝐻 = (Homa𝐶)
54homarcl 18050 . . . . . . 7 (𝐹 ∈ (𝑋𝐻𝑌) → 𝐶 ∈ Cat)
63, 5syl 17 . . . . . 6 (𝜑𝐶 ∈ Cat)
7 eqid 2726 . . . . . 6 (Id‘𝐶) = (Id‘𝐶)
84, 2homarcl2 18057 . . . . . . . 8 (𝐹 ∈ (𝑋𝐻𝑌) → (𝑋 ∈ (Base‘𝐶) ∧ 𝑌 ∈ (Base‘𝐶)))
93, 8syl 17 . . . . . . 7 (𝜑 → (𝑋 ∈ (Base‘𝐶) ∧ 𝑌 ∈ (Base‘𝐶)))
109simpld 493 . . . . . 6 (𝜑𝑋 ∈ (Base‘𝐶))
111, 2, 6, 7, 10ida2 18081 . . . . 5 (𝜑 → (2nd ‘( 1𝑋)) = ((Id‘𝐶)‘𝑋))
1211oveq2d 7440 . . . 4 (𝜑 → ((2nd𝐹)(⟨𝑋, 𝑋⟩(comp‘𝐶)𝑌)(2nd ‘( 1𝑋))) = ((2nd𝐹)(⟨𝑋, 𝑋⟩(comp‘𝐶)𝑌)((Id‘𝐶)‘𝑋)))
13 eqid 2726 . . . . 5 (Hom ‘𝐶) = (Hom ‘𝐶)
14 eqid 2726 . . . . 5 (comp‘𝐶) = (comp‘𝐶)
159simprd 494 . . . . 5 (𝜑𝑌 ∈ (Base‘𝐶))
164, 13homahom 18061 . . . . . 6 (𝐹 ∈ (𝑋𝐻𝑌) → (2nd𝐹) ∈ (𝑋(Hom ‘𝐶)𝑌))
173, 16syl 17 . . . . 5 (𝜑 → (2nd𝐹) ∈ (𝑋(Hom ‘𝐶)𝑌))
182, 13, 7, 6, 10, 14, 15, 17catrid 17697 . . . 4 (𝜑 → ((2nd𝐹)(⟨𝑋, 𝑋⟩(comp‘𝐶)𝑌)((Id‘𝐶)‘𝑋)) = (2nd𝐹))
1912, 18eqtrd 2766 . . 3 (𝜑 → ((2nd𝐹)(⟨𝑋, 𝑋⟩(comp‘𝐶)𝑌)(2nd ‘( 1𝑋))) = (2nd𝐹))
2019oteq3d 4893 . 2 (𝜑 → ⟨𝑋, 𝑌, ((2nd𝐹)(⟨𝑋, 𝑋⟩(comp‘𝐶)𝑌)(2nd ‘( 1𝑋)))⟩ = ⟨𝑋, 𝑌, (2nd𝐹)⟩)
21 arwlid.o . . 3 · = (compa𝐶)
221, 2, 6, 10, 4idahom 18082 . . 3 (𝜑 → ( 1𝑋) ∈ (𝑋𝐻𝑋))
2321, 4, 22, 3, 14coaval 18090 . 2 (𝜑 → (𝐹 · ( 1𝑋)) = ⟨𝑋, 𝑌, ((2nd𝐹)(⟨𝑋, 𝑋⟩(comp‘𝐶)𝑌)(2nd ‘( 1𝑋)))⟩)
244homadmcd 18064 . . 3 (𝐹 ∈ (𝑋𝐻𝑌) → 𝐹 = ⟨𝑋, 𝑌, (2nd𝐹)⟩)
253, 24syl 17 . 2 (𝜑𝐹 = ⟨𝑋, 𝑌, (2nd𝐹)⟩)
2620, 23, 253eqtr4d 2776 1 (𝜑 → (𝐹 · ( 1𝑋)) = 𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394   = wceq 1534  wcel 2099  cop 4639  cotp 4641  cfv 6554  (class class class)co 7424  2nd c2nd 8002  Basecbs 17213  Hom chom 17277  compcco 17278  Catccat 17677  Idccid 17678  Homachoma 18045  Idacida 18075  compaccoa 18076
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-rep 5290  ax-sep 5304  ax-nul 5311  ax-pow 5369  ax-pr 5433  ax-un 7746
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-rmo 3364  df-reu 3365  df-rab 3420  df-v 3464  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4326  df-if 4534  df-pw 4609  df-sn 4634  df-pr 4636  df-op 4640  df-ot 4642  df-uni 4914  df-iun 5003  df-br 5154  df-opab 5216  df-mpt 5237  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6506  df-fun 6556  df-fn 6557  df-f 6558  df-f1 6559  df-fo 6560  df-f1o 6561  df-fv 6562  df-riota 7380  df-ov 7427  df-oprab 7428  df-mpo 7429  df-1st 8003  df-2nd 8004  df-cat 17681  df-cid 17682  df-doma 18046  df-coda 18047  df-homa 18048  df-arw 18049  df-ida 18077  df-coa 18078
This theorem is referenced by: (None)
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