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Theorem arwrid 16494
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 2609 . . . . . 6 (Base‘𝐶) = (Base‘𝐶)
3 arwlid.f . . . . . . 7 (𝜑𝐹 ∈ (𝑋𝐻𝑌))
4 arwlid.h . . . . . . . 8 𝐻 = (Homa𝐶)
54homarcl 16449 . . . . . . 7 (𝐹 ∈ (𝑋𝐻𝑌) → 𝐶 ∈ Cat)
63, 5syl 17 . . . . . 6 (𝜑𝐶 ∈ Cat)
7 eqid 2609 . . . . . 6 (Id‘𝐶) = (Id‘𝐶)
84, 2homarcl2 16456 . . . . . . . 8 (𝐹 ∈ (𝑋𝐻𝑌) → (𝑋 ∈ (Base‘𝐶) ∧ 𝑌 ∈ (Base‘𝐶)))
93, 8syl 17 . . . . . . 7 (𝜑 → (𝑋 ∈ (Base‘𝐶) ∧ 𝑌 ∈ (Base‘𝐶)))
109simpld 473 . . . . . 6 (𝜑𝑋 ∈ (Base‘𝐶))
111, 2, 6, 7, 10ida2 16480 . . . . 5 (𝜑 → (2nd ‘( 1𝑋)) = ((Id‘𝐶)‘𝑋))
1211oveq2d 6542 . . . 4 (𝜑 → ((2nd𝐹)(⟨𝑋, 𝑋⟩(comp‘𝐶)𝑌)(2nd ‘( 1𝑋))) = ((2nd𝐹)(⟨𝑋, 𝑋⟩(comp‘𝐶)𝑌)((Id‘𝐶)‘𝑋)))
13 eqid 2609 . . . . 5 (Hom ‘𝐶) = (Hom ‘𝐶)
14 eqid 2609 . . . . 5 (comp‘𝐶) = (comp‘𝐶)
159simprd 477 . . . . 5 (𝜑𝑌 ∈ (Base‘𝐶))
164, 13homahom 16460 . . . . . 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 16116 . . . 4 (𝜑 → ((2nd𝐹)(⟨𝑋, 𝑋⟩(comp‘𝐶)𝑌)((Id‘𝐶)‘𝑋)) = (2nd𝐹))
1912, 18eqtrd 2643 . . 3 (𝜑 → ((2nd𝐹)(⟨𝑋, 𝑋⟩(comp‘𝐶)𝑌)(2nd ‘( 1𝑋))) = (2nd𝐹))
2019oteq3d 4348 . 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 16481 . . 3 (𝜑 → ( 1𝑋) ∈ (𝑋𝐻𝑋))
2321, 4, 22, 3, 14coaval 16489 . 2 (𝜑 → (𝐹 · ( 1𝑋)) = ⟨𝑋, 𝑌, ((2nd𝐹)(⟨𝑋, 𝑋⟩(comp‘𝐶)𝑌)(2nd ‘( 1𝑋)))⟩)
244homadmcd 16463 . . 3 (𝐹 ∈ (𝑋𝐻𝑌) → 𝐹 = ⟨𝑋, 𝑌, (2nd𝐹)⟩)
253, 24syl 17 . 2 (𝜑𝐹 = ⟨𝑋, 𝑌, (2nd𝐹)⟩)
2620, 23, 253eqtr4d 2653 1 (𝜑 → (𝐹 · ( 1𝑋)) = 𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1474  wcel 1976  cop 4130  cotp 4132  cfv 5789  (class class class)co 6526  2nd c2nd 7035  Basecbs 15643  Hom chom 15727  compcco 15728  Catccat 16096  Idccid 16097  Homachoma 16444  Idacida 16474  compaccoa 16475
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-rep 4693  ax-sep 4703  ax-nul 4711  ax-pow 4763  ax-pr 4827  ax-un 6824
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-ral 2900  df-rex 2901  df-reu 2902  df-rmo 2903  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-op 4131  df-ot 4133  df-uni 4367  df-iun 4451  df-br 4578  df-opab 4638  df-mpt 4639  df-id 4942  df-xp 5033  df-rel 5034  df-cnv 5035  df-co 5036  df-dm 5037  df-rn 5038  df-res 5039  df-ima 5040  df-iota 5753  df-fun 5791  df-fn 5792  df-f 5793  df-f1 5794  df-fo 5795  df-f1o 5796  df-fv 5797  df-riota 6488  df-ov 6529  df-oprab 6530  df-mpt2 6531  df-1st 7036  df-2nd 7037  df-cat 16100  df-cid 16101  df-doma 16445  df-coda 16446  df-homa 16447  df-arw 16448  df-ida 16476  df-coa 16477
This theorem is referenced by: (None)
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