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Theorem arwlid 16494
Description: Left 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
arwlid (𝜑 → (( 1𝑌) · 𝐹) = 𝐹)

Proof of Theorem arwlid
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 16450 . . . . . . 7 (𝐹 ∈ (𝑋𝐻𝑌) → 𝐶 ∈ Cat)
63, 5syl 17 . . . . . 6 (𝜑𝐶 ∈ Cat)
7 eqid 2610 . . . . . 6 (Id‘𝐶) = (Id‘𝐶)
84, 2homarcl2 16457 . . . . . . . 8 (𝐹 ∈ (𝑋𝐻𝑌) → (𝑋 ∈ (Base‘𝐶) ∧ 𝑌 ∈ (Base‘𝐶)))
93, 8syl 17 . . . . . . 7 (𝜑 → (𝑋 ∈ (Base‘𝐶) ∧ 𝑌 ∈ (Base‘𝐶)))
109simprd 478 . . . . . 6 (𝜑𝑌 ∈ (Base‘𝐶))
111, 2, 6, 7, 10ida2 16481 . . . . 5 (𝜑 → (2nd ‘( 1𝑌)) = ((Id‘𝐶)‘𝑌))
1211oveq1d 6542 . . . 4 (𝜑 → ((2nd ‘( 1𝑌))(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑌)(2nd𝐹)) = (((Id‘𝐶)‘𝑌)(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑌)(2nd𝐹)))
13 eqid 2610 . . . . 5 (Hom ‘𝐶) = (Hom ‘𝐶)
149simpld 474 . . . . 5 (𝜑𝑋 ∈ (Base‘𝐶))
15 eqid 2610 . . . . 5 (comp‘𝐶) = (comp‘𝐶)
164, 13homahom 16461 . . . . . 6 (𝐹 ∈ (𝑋𝐻𝑌) → (2nd𝐹) ∈ (𝑋(Hom ‘𝐶)𝑌))
173, 16syl 17 . . . . 5 (𝜑 → (2nd𝐹) ∈ (𝑋(Hom ‘𝐶)𝑌))
182, 13, 7, 6, 14, 15, 10, 17catlid 16116 . . . 4 (𝜑 → (((Id‘𝐶)‘𝑌)(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑌)(2nd𝐹)) = (2nd𝐹))
1912, 18eqtrd 2644 . . 3 (𝜑 → ((2nd ‘( 1𝑌))(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑌)(2nd𝐹)) = (2nd𝐹))
2019oteq3d 4349 . 2 (𝜑 → ⟨𝑋, 𝑌, ((2nd ‘( 1𝑌))(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑌)(2nd𝐹))⟩ = ⟨𝑋, 𝑌, (2nd𝐹)⟩)
21 arwlid.o . . 3 · = (compa𝐶)
221, 2, 6, 10, 4idahom 16482 . . 3 (𝜑 → ( 1𝑌) ∈ (𝑌𝐻𝑌))
2321, 4, 3, 22, 15coaval 16490 . 2 (𝜑 → (( 1𝑌) · 𝐹) = ⟨𝑋, 𝑌, ((2nd ‘( 1𝑌))(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑌)(2nd𝐹))⟩)
244homadmcd 16464 . . 3 (𝐹 ∈ (𝑋𝐻𝑌) → 𝐹 = ⟨𝑋, 𝑌, (2nd𝐹)⟩)
253, 24syl 17 . 2 (𝜑𝐹 = ⟨𝑋, 𝑌, (2nd𝐹)⟩)
2620, 23, 253eqtr4d 2654 1 (𝜑 → (( 1𝑌) · 𝐹) = 𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1475  wcel 1977  cop 4131  cotp 4133  cfv 5790  (class class class)co 6527  2nd c2nd 7036  Basecbs 15644  Hom chom 15728  compcco 15729  Catccat 16097  Idccid 16098  Homachoma 16445  Idacida 16475  compaccoa 16476
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 4694  ax-sep 4704  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6825
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 4368  df-iun 4452  df-br 4579  df-opab 4639  df-mpt 4640  df-id 4943  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-riota 6489  df-ov 6530  df-oprab 6531  df-mpt2 6532  df-1st 7037  df-2nd 7038  df-cat 16101  df-cid 16102  df-doma 16446  df-coda 16447  df-homa 16448  df-arw 16449  df-ida 16477  df-coa 16478
This theorem is referenced by: (None)
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