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Theorem arwrid 16546
 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 2610 . . . . . 6 (Base‘𝐶) = (Base‘𝐶)
3 arwlid.f . . . . . . 7 (𝜑𝐹 ∈ (𝑋𝐻𝑌))
4 arwlid.h . . . . . . . 8 𝐻 = (Homa𝐶)
54homarcl 16501 . . . . . . 7 (𝐹 ∈ (𝑋𝐻𝑌) → 𝐶 ∈ Cat)
63, 5syl 17 . . . . . 6 (𝜑𝐶 ∈ Cat)
7 eqid 2610 . . . . . 6 (Id‘𝐶) = (Id‘𝐶)
84, 2homarcl2 16508 . . . . . . . 8 (𝐹 ∈ (𝑋𝐻𝑌) → (𝑋 ∈ (Base‘𝐶) ∧ 𝑌 ∈ (Base‘𝐶)))
93, 8syl 17 . . . . . . 7 (𝜑 → (𝑋 ∈ (Base‘𝐶) ∧ 𝑌 ∈ (Base‘𝐶)))
109simpld 474 . . . . . 6 (𝜑𝑋 ∈ (Base‘𝐶))
111, 2, 6, 7, 10ida2 16532 . . . . 5 (𝜑 → (2nd ‘( 1𝑋)) = ((Id‘𝐶)‘𝑋))
1211oveq2d 6565 . . . 4 (𝜑 → ((2nd𝐹)(⟨𝑋, 𝑋⟩(comp‘𝐶)𝑌)(2nd ‘( 1𝑋))) = ((2nd𝐹)(⟨𝑋, 𝑋⟩(comp‘𝐶)𝑌)((Id‘𝐶)‘𝑋)))
13 eqid 2610 . . . . 5 (Hom ‘𝐶) = (Hom ‘𝐶)
14 eqid 2610 . . . . 5 (comp‘𝐶) = (comp‘𝐶)
159simprd 478 . . . . 5 (𝜑𝑌 ∈ (Base‘𝐶))
164, 13homahom 16512 . . . . . 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 16168 . . . 4 (𝜑 → ((2nd𝐹)(⟨𝑋, 𝑋⟩(comp‘𝐶)𝑌)((Id‘𝐶)‘𝑋)) = (2nd𝐹))
1912, 18eqtrd 2644 . . 3 (𝜑 → ((2nd𝐹)(⟨𝑋, 𝑋⟩(comp‘𝐶)𝑌)(2nd ‘( 1𝑋))) = (2nd𝐹))
2019oteq3d 4354 . 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 16533 . . 3 (𝜑 → ( 1𝑋) ∈ (𝑋𝐻𝑋))
2321, 4, 22, 3, 14coaval 16541 . 2 (𝜑 → (𝐹 · ( 1𝑋)) = ⟨𝑋, 𝑌, ((2nd𝐹)(⟨𝑋, 𝑋⟩(comp‘𝐶)𝑌)(2nd ‘( 1𝑋)))⟩)
244homadmcd 16515 . . 3 (𝐹 ∈ (𝑋𝐻𝑌) → 𝐹 = ⟨𝑋, 𝑌, (2nd𝐹)⟩)
253, 24syl 17 . 2 (𝜑𝐹 = ⟨𝑋, 𝑌, (2nd𝐹)⟩)
2620, 23, 253eqtr4d 2654 1 (𝜑 → (𝐹 · ( 1𝑋)) = 𝐹)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ⟨cop 4131  ⟨cotp 4133  ‘cfv 5804  (class class class)co 6549  2nd c2nd 7058  Basecbs 15695  Hom chom 15779  compcco 15780  Catccat 16148  Idccid 16149  Homachoma 16496  Idacida 16526  compaccoa 16527 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-ot 4134  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-1st 7059  df-2nd 7060  df-cat 16152  df-cid 16153  df-doma 16497  df-coda 16498  df-homa 16499  df-arw 16500  df-ida 16528  df-coa 16529 This theorem is referenced by: (None)
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