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Theorem catlid 16956
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 7166 . . 3 (𝑓 = 𝐹 → (( 1𝑌)(⟨𝑋, 𝑌· 𝑌)𝑓) = (( 1𝑌)(⟨𝑋, 𝑌· 𝑌)𝐹))
2 id 22 . . 3 (𝑓 = 𝐹𝑓 = 𝐹)
31, 2eqeq12d 2839 . 2 (𝑓 = 𝐹 → ((( 1𝑌)(⟨𝑋, 𝑌· 𝑌)𝑓) = 𝑓 ↔ (( 1𝑌)(⟨𝑋, 𝑌· 𝑌)𝐹) = 𝐹))
4 oveq1 7165 . . . 4 (𝑥 = 𝑋 → (𝑥𝐻𝑌) = (𝑋𝐻𝑌))
5 opeq1 4805 . . . . . . 7 (𝑥 = 𝑋 → ⟨𝑥, 𝑌⟩ = ⟨𝑋, 𝑌⟩)
65oveq1d 7173 . . . . . 6 (𝑥 = 𝑋 → (⟨𝑥, 𝑌· 𝑌) = (⟨𝑋, 𝑌· 𝑌))
76oveqd 7175 . . . . 5 (𝑥 = 𝑋 → (( 1𝑌)(⟨𝑥, 𝑌· 𝑌)𝑓) = (( 1𝑌)(⟨𝑋, 𝑌· 𝑌)𝑓))
87eqeq1d 2825 . . . 4 (𝑥 = 𝑋 → ((( 1𝑌)(⟨𝑥, 𝑌· 𝑌)𝑓) = 𝑓 ↔ (( 1𝑌)(⟨𝑋, 𝑌· 𝑌)𝑓) = 𝑓))
94, 8raleqbidv 3403 . . 3 (𝑥 = 𝑋 → (∀𝑓 ∈ (𝑥𝐻𝑌)(( 1𝑌)(⟨𝑥, 𝑌· 𝑌)𝑓) = 𝑓 ↔ ∀𝑓 ∈ (𝑋𝐻𝑌)(( 1𝑌)(⟨𝑋, 𝑌· 𝑌)𝑓) = 𝑓))
10 simpl 485 . . . . . . . 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 3162 . . . . . . 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 4055 . . . . 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 16950 . . . . . 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 16948 . . . . . . 7 (𝜑 → ∃!𝑔 ∈ (𝑌𝐻𝑌)∀𝑥𝐵 (∀𝑓 ∈ (𝑥𝐻𝑌)(𝑔(⟨𝑥, 𝑌· 𝑌)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑌𝐻𝑥)(𝑓(⟨𝑌, 𝑌· 𝑥)𝑔) = 𝑓))
22 riotacl2 7132 . . . . . . 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 2915 . . . . 5 (𝜑 → ( 1𝑌) ∈ {𝑔 ∈ (𝑌𝐻𝑌) ∣ ∀𝑥𝐵 (∀𝑓 ∈ (𝑥𝐻𝑌)(𝑔(⟨𝑥, 𝑌· 𝑌)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑌𝐻𝑥)(𝑓(⟨𝑌, 𝑌· 𝑥)𝑔) = 𝑓)})
2513, 24sseldi 3967 . . . 4 (𝜑 → ( 1𝑌) ∈ {𝑔 ∈ (𝑌𝐻𝑌) ∣ ∀𝑥𝐵𝑓 ∈ (𝑥𝐻𝑌)(𝑔(⟨𝑥, 𝑌· 𝑌)𝑓) = 𝑓})
26 oveq1 7165 . . . . . . . 8 (𝑔 = ( 1𝑌) → (𝑔(⟨𝑥, 𝑌· 𝑌)𝑓) = (( 1𝑌)(⟨𝑥, 𝑌· 𝑌)𝑓))
2726eqeq1d 2825 . . . . . . 7 (𝑔 = ( 1𝑌) → ((𝑔(⟨𝑥, 𝑌· 𝑌)𝑓) = 𝑓 ↔ (( 1𝑌)(⟨𝑥, 𝑌· 𝑌)𝑓) = 𝑓))
28272ralbidv 3201 . . . . . 6 (𝑔 = ( 1𝑌) → (∀𝑥𝐵𝑓 ∈ (𝑥𝐻𝑌)(𝑔(⟨𝑥, 𝑌· 𝑌)𝑓) = 𝑓 ↔ ∀𝑥𝐵𝑓 ∈ (𝑥𝐻𝑌)(( 1𝑌)(⟨𝑥, 𝑌· 𝑌)𝑓) = 𝑓))
2928elrab 3682 . . . . 5 (( 1𝑌) ∈ {𝑔 ∈ (𝑌𝐻𝑌) ∣ ∀𝑥𝐵𝑓 ∈ (𝑥𝐻𝑌)(𝑔(⟨𝑥, 𝑌· 𝑌)𝑓) = 𝑓} ↔ (( 1𝑌) ∈ (𝑌𝐻𝑌) ∧ ∀𝑥𝐵𝑓 ∈ (𝑥𝐻𝑌)(( 1𝑌)(⟨𝑥, 𝑌· 𝑌)𝑓) = 𝑓))
3029simprbi 499 . . . 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 3627 . 2 (𝜑 → ∀𝑓 ∈ (𝑋𝐻𝑌)(( 1𝑌)(⟨𝑋, 𝑌· 𝑌)𝑓) = 𝑓)
34 catlid.f . 2 (𝜑𝐹 ∈ (𝑋𝐻𝑌))
353, 33, 34rspcdva 3627 1 (𝜑 → (( 1𝑌)(⟨𝑋, 𝑌· 𝑌)𝐹) = 𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 398   = wceq 1537  wcel 2114  wral 3140  ∃!wreu 3142  {crab 3144  cop 4575  cfv 6357  crio 7115  (class class class)co 7158  Basecbs 16485  Hom chom 16578  compcco 16579  Catccat 16937  Idccid 16938
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-rep 5192  ax-sep 5205  ax-nul 5212  ax-pr 5332
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-ral 3145  df-rex 3146  df-reu 3147  df-rmo 3148  df-rab 3149  df-v 3498  df-sbc 3775  df-csb 3886  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4841  df-iun 4923  df-br 5069  df-opab 5131  df-mpt 5149  df-id 5462  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-res 5569  df-ima 5570  df-iota 6316  df-fun 6359  df-fn 6360  df-f 6361  df-f1 6362  df-fo 6363  df-f1o 6364  df-fv 6365  df-riota 7116  df-ov 7161  df-cat 16941  df-cid 16942
This theorem is referenced by:  oppccatid  16991  sectcan  17027  sectco  17028  sectmon  17054  monsect  17055  sectid  17058  invisoinvl  17062  subccatid  17118  fucidcl  17237  fuclid  17238  invfuc  17246  arwlid  17334  xpccatid  17440  evlfcl  17474  curf1cl  17480  curf2cl  17483  curfcl  17484  curfuncf  17490  uncfcurf  17491  hofcl  17511  yon12  17517  yon2  17518  yonedalem3b  17531  yonedainv  17533  bj-endmnd  34601
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