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Theorem arwlid 16929
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 2771 . . . . . 6 (Base‘𝐶) = (Base‘𝐶)
3 arwlid.f . . . . . . 7 (𝜑𝐹 ∈ (𝑋𝐻𝑌))
4 arwlid.h . . . . . . . 8 𝐻 = (Homa𝐶)
54homarcl 16885 . . . . . . 7 (𝐹 ∈ (𝑋𝐻𝑌) → 𝐶 ∈ Cat)
63, 5syl 17 . . . . . 6 (𝜑𝐶 ∈ Cat)
7 eqid 2771 . . . . . 6 (Id‘𝐶) = (Id‘𝐶)
84, 2homarcl2 16892 . . . . . . . 8 (𝐹 ∈ (𝑋𝐻𝑌) → (𝑋 ∈ (Base‘𝐶) ∧ 𝑌 ∈ (Base‘𝐶)))
93, 8syl 17 . . . . . . 7 (𝜑 → (𝑋 ∈ (Base‘𝐶) ∧ 𝑌 ∈ (Base‘𝐶)))
109simprd 483 . . . . . 6 (𝜑𝑌 ∈ (Base‘𝐶))
111, 2, 6, 7, 10ida2 16916 . . . . 5 (𝜑 → (2nd ‘( 1𝑌)) = ((Id‘𝐶)‘𝑌))
1211oveq1d 6811 . . . 4 (𝜑 → ((2nd ‘( 1𝑌))(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑌)(2nd𝐹)) = (((Id‘𝐶)‘𝑌)(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑌)(2nd𝐹)))
13 eqid 2771 . . . . 5 (Hom ‘𝐶) = (Hom ‘𝐶)
149simpld 482 . . . . 5 (𝜑𝑋 ∈ (Base‘𝐶))
15 eqid 2771 . . . . 5 (comp‘𝐶) = (comp‘𝐶)
164, 13homahom 16896 . . . . . 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 16551 . . . 4 (𝜑 → (((Id‘𝐶)‘𝑌)(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑌)(2nd𝐹)) = (2nd𝐹))
1912, 18eqtrd 2805 . . 3 (𝜑 → ((2nd ‘( 1𝑌))(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑌)(2nd𝐹)) = (2nd𝐹))
2019oteq3d 4554 . 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 16917 . . 3 (𝜑 → ( 1𝑌) ∈ (𝑌𝐻𝑌))
2321, 4, 3, 22, 15coaval 16925 . 2 (𝜑 → (( 1𝑌) · 𝐹) = ⟨𝑋, 𝑌, ((2nd ‘( 1𝑌))(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑌)(2nd𝐹))⟩)
244homadmcd 16899 . . 3 (𝐹 ∈ (𝑋𝐻𝑌) → 𝐹 = ⟨𝑋, 𝑌, (2nd𝐹)⟩)
253, 24syl 17 . 2 (𝜑𝐹 = ⟨𝑋, 𝑌, (2nd𝐹)⟩)
2620, 23, 253eqtr4d 2815 1 (𝜑 → (( 1𝑌) · 𝐹) = 𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1631  wcel 2145  cop 4323  cotp 4325  cfv 6030  (class class class)co 6796  2nd c2nd 7318  Basecbs 16064  Hom chom 16160  compcco 16161  Catccat 16532  Idccid 16533  Homachoma 16880  Idacida 16910  compaccoa 16911
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4905  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035  ax-un 7100
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-op 4324  df-ot 4326  df-uni 4576  df-iun 4657  df-br 4788  df-opab 4848  df-mpt 4865  df-id 5158  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-iota 5993  df-fun 6032  df-fn 6033  df-f 6034  df-f1 6035  df-fo 6036  df-f1o 6037  df-fv 6038  df-riota 6757  df-ov 6799  df-oprab 6800  df-mpt2 6801  df-1st 7319  df-2nd 7320  df-cat 16536  df-cid 16537  df-doma 16881  df-coda 16882  df-homa 16883  df-arw 16884  df-ida 16912  df-coa 16913
This theorem is referenced by: (None)
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