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Theorem catlid 16655
Description: Left identity property of an identity arrow. (Contributed by Mario Carneiro, 2-Jan-2017.)
Hypotheses
Ref Expression
catidcl.b 𝐵 = (Base‘𝐶)
catidcl.h 𝐻 = (Hom ‘𝐶)
catidcl.i 1 = (Id‘𝐶)
catidcl.c (𝜑𝐶 ∈ Cat)
catidcl.x (𝜑𝑋𝐵)
catlid.o · = (comp‘𝐶)
catlid.y (𝜑𝑌𝐵)
catlid.f (𝜑𝐹 ∈ (𝑋𝐻𝑌))
Assertion
Ref Expression
catlid (𝜑 → (( 1𝑌)(⟨𝑋, 𝑌· 𝑌)𝐹) = 𝐹)

Proof of Theorem catlid
Dummy variables 𝑓 𝑔 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq2 6884 . . 3 (𝑓 = 𝐹 → (( 1𝑌)(⟨𝑋, 𝑌· 𝑌)𝑓) = (( 1𝑌)(⟨𝑋, 𝑌· 𝑌)𝐹))
2 id 22 . . 3 (𝑓 = 𝐹𝑓 = 𝐹)
31, 2eqeq12d 2812 . 2 (𝑓 = 𝐹 → ((( 1𝑌)(⟨𝑋, 𝑌· 𝑌)𝑓) = 𝑓 ↔ (( 1𝑌)(⟨𝑋, 𝑌· 𝑌)𝐹) = 𝐹))
4 oveq1 6883 . . . 4 (𝑥 = 𝑋 → (𝑥𝐻𝑌) = (𝑋𝐻𝑌))
5 opeq1 4591 . . . . . . 7 (𝑥 = 𝑋 → ⟨𝑥, 𝑌⟩ = ⟨𝑋, 𝑌⟩)
65oveq1d 6891 . . . . . 6 (𝑥 = 𝑋 → (⟨𝑥, 𝑌· 𝑌) = (⟨𝑋, 𝑌· 𝑌))
76oveqd 6893 . . . . 5 (𝑥 = 𝑋 → (( 1𝑌)(⟨𝑥, 𝑌· 𝑌)𝑓) = (( 1𝑌)(⟨𝑋, 𝑌· 𝑌)𝑓))
87eqeq1d 2799 . . . 4 (𝑥 = 𝑋 → ((( 1𝑌)(⟨𝑥, 𝑌· 𝑌)𝑓) = 𝑓 ↔ (( 1𝑌)(⟨𝑋, 𝑌· 𝑌)𝑓) = 𝑓))
94, 8raleqbidv 3333 . . 3 (𝑥 = 𝑋 → (∀𝑓 ∈ (𝑥𝐻𝑌)(( 1𝑌)(⟨𝑥, 𝑌· 𝑌)𝑓) = 𝑓 ↔ ∀𝑓 ∈ (𝑋𝐻𝑌)(( 1𝑌)(⟨𝑋, 𝑌· 𝑌)𝑓) = 𝑓))
10 simpl 475 . . . . . . . 8 ((∀𝑓 ∈ (𝑥𝐻𝑌)(𝑔(⟨𝑥, 𝑌· 𝑌)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑌𝐻𝑥)(𝑓(⟨𝑌, 𝑌· 𝑥)𝑔) = 𝑓) → ∀𝑓 ∈ (𝑥𝐻𝑌)(𝑔(⟨𝑥, 𝑌· 𝑌)𝑓) = 𝑓)
1110ralimi 3131 . . . . . . 7 (∀𝑥𝐵 (∀𝑓 ∈ (𝑥𝐻𝑌)(𝑔(⟨𝑥, 𝑌· 𝑌)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑌𝐻𝑥)(𝑓(⟨𝑌, 𝑌· 𝑥)𝑔) = 𝑓) → ∀𝑥𝐵𝑓 ∈ (𝑥𝐻𝑌)(𝑔(⟨𝑥, 𝑌· 𝑌)𝑓) = 𝑓)
1211a1i 11 . . . . . 6 (𝑔 ∈ (𝑌𝐻𝑌) → (∀𝑥𝐵 (∀𝑓 ∈ (𝑥𝐻𝑌)(𝑔(⟨𝑥, 𝑌· 𝑌)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑌𝐻𝑥)(𝑓(⟨𝑌, 𝑌· 𝑥)𝑔) = 𝑓) → ∀𝑥𝐵𝑓 ∈ (𝑥𝐻𝑌)(𝑔(⟨𝑥, 𝑌· 𝑌)𝑓) = 𝑓))
1312ss2rabi 3878 . . . . 5 {𝑔 ∈ (𝑌𝐻𝑌) ∣ ∀𝑥𝐵 (∀𝑓 ∈ (𝑥𝐻𝑌)(𝑔(⟨𝑥, 𝑌· 𝑌)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑌𝐻𝑥)(𝑓(⟨𝑌, 𝑌· 𝑥)𝑔) = 𝑓)} ⊆ {𝑔 ∈ (𝑌𝐻𝑌) ∣ ∀𝑥𝐵𝑓 ∈ (𝑥𝐻𝑌)(𝑔(⟨𝑥, 𝑌· 𝑌)𝑓) = 𝑓}
14 catidcl.b . . . . . . 7 𝐵 = (Base‘𝐶)
15 catidcl.h . . . . . . 7 𝐻 = (Hom ‘𝐶)
16 catlid.o . . . . . . 7 · = (comp‘𝐶)
17 catidcl.c . . . . . . 7 (𝜑𝐶 ∈ Cat)
18 catidcl.i . . . . . . 7 1 = (Id‘𝐶)
19 catlid.y . . . . . . 7 (𝜑𝑌𝐵)
2014, 15, 16, 17, 18, 19cidval 16649 . . . . . 6 (𝜑 → ( 1𝑌) = (𝑔 ∈ (𝑌𝐻𝑌)∀𝑥𝐵 (∀𝑓 ∈ (𝑥𝐻𝑌)(𝑔(⟨𝑥, 𝑌· 𝑌)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑌𝐻𝑥)(𝑓(⟨𝑌, 𝑌· 𝑥)𝑔) = 𝑓)))
2114, 15, 16, 17, 19catideu 16647 . . . . . . 7 (𝜑 → ∃!𝑔 ∈ (𝑌𝐻𝑌)∀𝑥𝐵 (∀𝑓 ∈ (𝑥𝐻𝑌)(𝑔(⟨𝑥, 𝑌· 𝑌)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑌𝐻𝑥)(𝑓(⟨𝑌, 𝑌· 𝑥)𝑔) = 𝑓))
22 riotacl2 6850 . . . . . . 7 (∃!𝑔 ∈ (𝑌𝐻𝑌)∀𝑥𝐵 (∀𝑓 ∈ (𝑥𝐻𝑌)(𝑔(⟨𝑥, 𝑌· 𝑌)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑌𝐻𝑥)(𝑓(⟨𝑌, 𝑌· 𝑥)𝑔) = 𝑓) → (𝑔 ∈ (𝑌𝐻𝑌)∀𝑥𝐵 (∀𝑓 ∈ (𝑥𝐻𝑌)(𝑔(⟨𝑥, 𝑌· 𝑌)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑌𝐻𝑥)(𝑓(⟨𝑌, 𝑌· 𝑥)𝑔) = 𝑓)) ∈ {𝑔 ∈ (𝑌𝐻𝑌) ∣ ∀𝑥𝐵 (∀𝑓 ∈ (𝑥𝐻𝑌)(𝑔(⟨𝑥, 𝑌· 𝑌)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑌𝐻𝑥)(𝑓(⟨𝑌, 𝑌· 𝑥)𝑔) = 𝑓)})
2321, 22syl 17 . . . . . 6 (𝜑 → (𝑔 ∈ (𝑌𝐻𝑌)∀𝑥𝐵 (∀𝑓 ∈ (𝑥𝐻𝑌)(𝑔(⟨𝑥, 𝑌· 𝑌)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑌𝐻𝑥)(𝑓(⟨𝑌, 𝑌· 𝑥)𝑔) = 𝑓)) ∈ {𝑔 ∈ (𝑌𝐻𝑌) ∣ ∀𝑥𝐵 (∀𝑓 ∈ (𝑥𝐻𝑌)(𝑔(⟨𝑥, 𝑌· 𝑌)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑌𝐻𝑥)(𝑓(⟨𝑌, 𝑌· 𝑥)𝑔) = 𝑓)})
2420, 23eqeltrd 2876 . . . . 5 (𝜑 → ( 1𝑌) ∈ {𝑔 ∈ (𝑌𝐻𝑌) ∣ ∀𝑥𝐵 (∀𝑓 ∈ (𝑥𝐻𝑌)(𝑔(⟨𝑥, 𝑌· 𝑌)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑌𝐻𝑥)(𝑓(⟨𝑌, 𝑌· 𝑥)𝑔) = 𝑓)})
2513, 24sseldi 3794 . . . 4 (𝜑 → ( 1𝑌) ∈ {𝑔 ∈ (𝑌𝐻𝑌) ∣ ∀𝑥𝐵𝑓 ∈ (𝑥𝐻𝑌)(𝑔(⟨𝑥, 𝑌· 𝑌)𝑓) = 𝑓})
26 oveq1 6883 . . . . . . . 8 (𝑔 = ( 1𝑌) → (𝑔(⟨𝑥, 𝑌· 𝑌)𝑓) = (( 1𝑌)(⟨𝑥, 𝑌· 𝑌)𝑓))
2726eqeq1d 2799 . . . . . . 7 (𝑔 = ( 1𝑌) → ((𝑔(⟨𝑥, 𝑌· 𝑌)𝑓) = 𝑓 ↔ (( 1𝑌)(⟨𝑥, 𝑌· 𝑌)𝑓) = 𝑓))
28272ralbidv 3168 . . . . . 6 (𝑔 = ( 1𝑌) → (∀𝑥𝐵𝑓 ∈ (𝑥𝐻𝑌)(𝑔(⟨𝑥, 𝑌· 𝑌)𝑓) = 𝑓 ↔ ∀𝑥𝐵𝑓 ∈ (𝑥𝐻𝑌)(( 1𝑌)(⟨𝑥, 𝑌· 𝑌)𝑓) = 𝑓))
2928elrab 3554 . . . . 5 (( 1𝑌) ∈ {𝑔 ∈ (𝑌𝐻𝑌) ∣ ∀𝑥𝐵𝑓 ∈ (𝑥𝐻𝑌)(𝑔(⟨𝑥, 𝑌· 𝑌)𝑓) = 𝑓} ↔ (( 1𝑌) ∈ (𝑌𝐻𝑌) ∧ ∀𝑥𝐵𝑓 ∈ (𝑥𝐻𝑌)(( 1𝑌)(⟨𝑥, 𝑌· 𝑌)𝑓) = 𝑓))
3029simprbi 491 . . . 4 (( 1𝑌) ∈ {𝑔 ∈ (𝑌𝐻𝑌) ∣ ∀𝑥𝐵𝑓 ∈ (𝑥𝐻𝑌)(𝑔(⟨𝑥, 𝑌· 𝑌)𝑓) = 𝑓} → ∀𝑥𝐵𝑓 ∈ (𝑥𝐻𝑌)(( 1𝑌)(⟨𝑥, 𝑌· 𝑌)𝑓) = 𝑓)
3125, 30syl 17 . . 3 (𝜑 → ∀𝑥𝐵𝑓 ∈ (𝑥𝐻𝑌)(( 1𝑌)(⟨𝑥, 𝑌· 𝑌)𝑓) = 𝑓)
32 catidcl.x . . 3 (𝜑𝑋𝐵)
339, 31, 32rspcdva 3501 . 2 (𝜑 → ∀𝑓 ∈ (𝑋𝐻𝑌)(( 1𝑌)(⟨𝑋, 𝑌· 𝑌)𝑓) = 𝑓)
34 catlid.f . 2 (𝜑𝐹 ∈ (𝑋𝐻𝑌))
353, 33, 34rspcdva 3501 1 (𝜑 → (( 1𝑌)(⟨𝑋, 𝑌· 𝑌)𝐹) = 𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 385   = wceq 1653  wcel 2157  wral 3087  ∃!wreu 3089  {crab 3091  cop 4372  cfv 6099  crio 6836  (class class class)co 6876  Basecbs 16181  Hom chom 16275  compcco 16276  Catccat 16636  Idccid 16637
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2354  ax-ext 2775  ax-rep 4962  ax-sep 4973  ax-nul 4981  ax-pr 5095
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2590  df-eu 2607  df-clab 2784  df-cleq 2790  df-clel 2793  df-nfc 2928  df-ne 2970  df-ral 3092  df-rex 3093  df-reu 3094  df-rmo 3095  df-rab 3096  df-v 3385  df-sbc 3632  df-csb 3727  df-dif 3770  df-un 3772  df-in 3774  df-ss 3781  df-nul 4114  df-if 4276  df-sn 4367  df-pr 4369  df-op 4373  df-uni 4627  df-iun 4710  df-br 4842  df-opab 4904  df-mpt 4921  df-id 5218  df-xp 5316  df-rel 5317  df-cnv 5318  df-co 5319  df-dm 5320  df-rn 5321  df-res 5322  df-ima 5323  df-iota 6062  df-fun 6101  df-fn 6102  df-f 6103  df-f1 6104  df-fo 6105  df-f1o 6106  df-fv 6107  df-riota 6837  df-ov 6879  df-cat 16640  df-cid 16641
This theorem is referenced by:  oppccatid  16690  sectcan  16726  sectco  16727  sectmon  16753  monsect  16754  sectid  16757  invisoinvl  16761  subccatid  16817  fucidcl  16936  fuclid  16937  invfuc  16945  arwlid  17033  xpccatid  17140  evlfcl  17174  curf1cl  17180  curf2cl  17183  curfcl  17184  curfuncf  17190  uncfcurf  17191  hofcl  17211  yon12  17217  yon2  17218  yonedalem3b  17231  yonedainv  17233
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