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Theorem arwlid 18126
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 2735 . . . . . 6 (Base‘𝐶) = (Base‘𝐶)
3 arwlid.f . . . . . . 7 (𝜑𝐹 ∈ (𝑋𝐻𝑌))
4 arwlid.h . . . . . . . 8 𝐻 = (Homa𝐶)
54homarcl 18082 . . . . . . 7 (𝐹 ∈ (𝑋𝐻𝑌) → 𝐶 ∈ Cat)
63, 5syl 17 . . . . . 6 (𝜑𝐶 ∈ Cat)
7 eqid 2735 . . . . . 6 (Id‘𝐶) = (Id‘𝐶)
84, 2homarcl2 18089 . . . . . . . 8 (𝐹 ∈ (𝑋𝐻𝑌) → (𝑋 ∈ (Base‘𝐶) ∧ 𝑌 ∈ (Base‘𝐶)))
93, 8syl 17 . . . . . . 7 (𝜑 → (𝑋 ∈ (Base‘𝐶) ∧ 𝑌 ∈ (Base‘𝐶)))
109simprd 495 . . . . . 6 (𝜑𝑌 ∈ (Base‘𝐶))
111, 2, 6, 7, 10ida2 18113 . . . . 5 (𝜑 → (2nd ‘( 1𝑌)) = ((Id‘𝐶)‘𝑌))
1211oveq1d 7446 . . . 4 (𝜑 → ((2nd ‘( 1𝑌))(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑌)(2nd𝐹)) = (((Id‘𝐶)‘𝑌)(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑌)(2nd𝐹)))
13 eqid 2735 . . . . 5 (Hom ‘𝐶) = (Hom ‘𝐶)
149simpld 494 . . . . 5 (𝜑𝑋 ∈ (Base‘𝐶))
15 eqid 2735 . . . . 5 (comp‘𝐶) = (comp‘𝐶)
164, 13homahom 18093 . . . . . 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 17728 . . . 4 (𝜑 → (((Id‘𝐶)‘𝑌)(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑌)(2nd𝐹)) = (2nd𝐹))
1912, 18eqtrd 2775 . . 3 (𝜑 → ((2nd ‘( 1𝑌))(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑌)(2nd𝐹)) = (2nd𝐹))
2019oteq3d 4892 . 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 18114 . . 3 (𝜑 → ( 1𝑌) ∈ (𝑌𝐻𝑌))
2321, 4, 3, 22, 15coaval 18122 . 2 (𝜑 → (( 1𝑌) · 𝐹) = ⟨𝑋, 𝑌, ((2nd ‘( 1𝑌))(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑌)(2nd𝐹))⟩)
244homadmcd 18096 . . 3 (𝐹 ∈ (𝑋𝐻𝑌) → 𝐹 = ⟨𝑋, 𝑌, (2nd𝐹)⟩)
253, 24syl 17 . 2 (𝜑𝐹 = ⟨𝑋, 𝑌, (2nd𝐹)⟩)
2620, 23, 253eqtr4d 2785 1 (𝜑 → (( 1𝑌) · 𝐹) = 𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wcel 2106  cop 4637  cotp 4639  cfv 6563  (class class class)co 7431  2nd c2nd 8012  Basecbs 17245  Hom chom 17309  compcco 17310  Catccat 17709  Idccid 17710  Homachoma 18077  Idacida 18107  compaccoa 18108
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-ot 4640  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8013  df-2nd 8014  df-cat 17713  df-cid 17714  df-doma 18078  df-coda 18079  df-homa 18080  df-arw 18081  df-ida 18109  df-coa 18110
This theorem is referenced by: (None)
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