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Theorem arwlid 17328
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 2801 . . . . . 6 (Base‘𝐶) = (Base‘𝐶)
3 arwlid.f . . . . . . 7 (𝜑𝐹 ∈ (𝑋𝐻𝑌))
4 arwlid.h . . . . . . . 8 𝐻 = (Homa𝐶)
54homarcl 17284 . . . . . . 7 (𝐹 ∈ (𝑋𝐻𝑌) → 𝐶 ∈ Cat)
63, 5syl 17 . . . . . 6 (𝜑𝐶 ∈ Cat)
7 eqid 2801 . . . . . 6 (Id‘𝐶) = (Id‘𝐶)
84, 2homarcl2 17291 . . . . . . . 8 (𝐹 ∈ (𝑋𝐻𝑌) → (𝑋 ∈ (Base‘𝐶) ∧ 𝑌 ∈ (Base‘𝐶)))
93, 8syl 17 . . . . . . 7 (𝜑 → (𝑋 ∈ (Base‘𝐶) ∧ 𝑌 ∈ (Base‘𝐶)))
109simprd 499 . . . . . 6 (𝜑𝑌 ∈ (Base‘𝐶))
111, 2, 6, 7, 10ida2 17315 . . . . 5 (𝜑 → (2nd ‘( 1𝑌)) = ((Id‘𝐶)‘𝑌))
1211oveq1d 7154 . . . 4 (𝜑 → ((2nd ‘( 1𝑌))(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑌)(2nd𝐹)) = (((Id‘𝐶)‘𝑌)(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑌)(2nd𝐹)))
13 eqid 2801 . . . . 5 (Hom ‘𝐶) = (Hom ‘𝐶)
149simpld 498 . . . . 5 (𝜑𝑋 ∈ (Base‘𝐶))
15 eqid 2801 . . . . 5 (comp‘𝐶) = (comp‘𝐶)
164, 13homahom 17295 . . . . . 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 16950 . . . 4 (𝜑 → (((Id‘𝐶)‘𝑌)(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑌)(2nd𝐹)) = (2nd𝐹))
1912, 18eqtrd 2836 . . 3 (𝜑 → ((2nd ‘( 1𝑌))(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑌)(2nd𝐹)) = (2nd𝐹))
2019oteq3d 4782 . 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 17316 . . 3 (𝜑 → ( 1𝑌) ∈ (𝑌𝐻𝑌))
2321, 4, 3, 22, 15coaval 17324 . 2 (𝜑 → (( 1𝑌) · 𝐹) = ⟨𝑋, 𝑌, ((2nd ‘( 1𝑌))(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑌)(2nd𝐹))⟩)
244homadmcd 17298 . . 3 (𝐹 ∈ (𝑋𝐻𝑌) → 𝐹 = ⟨𝑋, 𝑌, (2nd𝐹)⟩)
253, 24syl 17 . 2 (𝜑𝐹 = ⟨𝑋, 𝑌, (2nd𝐹)⟩)
2620, 23, 253eqtr4d 2846 1 (𝜑 → (( 1𝑌) · 𝐹) = 𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2112  cop 4534  cotp 4536  cfv 6328  (class class class)co 7139  2nd c2nd 7674  Basecbs 16479  Hom chom 16572  compcco 16573  Catccat 16931  Idccid 16932  Homachoma 17279  Idacida 17309  compaccoa 17310
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-reu 3116  df-rmo 3117  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-op 4535  df-ot 4537  df-uni 4804  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-riota 7097  df-ov 7142  df-oprab 7143  df-mpo 7144  df-1st 7675  df-2nd 7676  df-cat 16935  df-cid 16936  df-doma 17280  df-coda 17281  df-homa 17282  df-arw 17283  df-ida 17311  df-coa 17312
This theorem is referenced by: (None)
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