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Theorem catrid 16813
Description: Right 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
catrid (𝜑 → (𝐹(⟨𝑋, 𝑋· 𝑌)( 1𝑋)) = 𝐹)

Proof of Theorem catrid
Dummy variables 𝑓 𝑔 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq1 6983 . . 3 (𝑓 = 𝐹 → (𝑓(⟨𝑋, 𝑋· 𝑌)( 1𝑋)) = (𝐹(⟨𝑋, 𝑋· 𝑌)( 1𝑋)))
2 id 22 . . 3 (𝑓 = 𝐹𝑓 = 𝐹)
31, 2eqeq12d 2793 . 2 (𝑓 = 𝐹 → ((𝑓(⟨𝑋, 𝑋· 𝑌)( 1𝑋)) = 𝑓 ↔ (𝐹(⟨𝑋, 𝑋· 𝑌)( 1𝑋)) = 𝐹))
4 oveq2 6984 . . . 4 (𝑦 = 𝑌 → (𝑋𝐻𝑦) = (𝑋𝐻𝑌))
5 oveq2 6984 . . . . . 6 (𝑦 = 𝑌 → (⟨𝑋, 𝑋· 𝑦) = (⟨𝑋, 𝑋· 𝑌))
65oveqd 6993 . . . . 5 (𝑦 = 𝑌 → (𝑓(⟨𝑋, 𝑋· 𝑦)( 1𝑋)) = (𝑓(⟨𝑋, 𝑋· 𝑌)( 1𝑋)))
76eqeq1d 2780 . . . 4 (𝑦 = 𝑌 → ((𝑓(⟨𝑋, 𝑋· 𝑦)( 1𝑋)) = 𝑓 ↔ (𝑓(⟨𝑋, 𝑋· 𝑌)( 1𝑋)) = 𝑓))
84, 7raleqbidv 3341 . . 3 (𝑦 = 𝑌 → (∀𝑓 ∈ (𝑋𝐻𝑦)(𝑓(⟨𝑋, 𝑋· 𝑦)( 1𝑋)) = 𝑓 ↔ ∀𝑓 ∈ (𝑋𝐻𝑌)(𝑓(⟨𝑋, 𝑋· 𝑌)( 1𝑋)) = 𝑓))
9 simpr 477 . . . . . . . 8 ((∀𝑓 ∈ (𝑦𝐻𝑋)(𝑔(⟨𝑦, 𝑋· 𝑋)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑋𝐻𝑦)(𝑓(⟨𝑋, 𝑋· 𝑦)𝑔) = 𝑓) → ∀𝑓 ∈ (𝑋𝐻𝑦)(𝑓(⟨𝑋, 𝑋· 𝑦)𝑔) = 𝑓)
109ralimi 3110 . . . . . . 7 (∀𝑦𝐵 (∀𝑓 ∈ (𝑦𝐻𝑋)(𝑔(⟨𝑦, 𝑋· 𝑋)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑋𝐻𝑦)(𝑓(⟨𝑋, 𝑋· 𝑦)𝑔) = 𝑓) → ∀𝑦𝐵𝑓 ∈ (𝑋𝐻𝑦)(𝑓(⟨𝑋, 𝑋· 𝑦)𝑔) = 𝑓)
1110a1i 11 . . . . . 6 (𝑔 ∈ (𝑋𝐻𝑋) → (∀𝑦𝐵 (∀𝑓 ∈ (𝑦𝐻𝑋)(𝑔(⟨𝑦, 𝑋· 𝑋)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑋𝐻𝑦)(𝑓(⟨𝑋, 𝑋· 𝑦)𝑔) = 𝑓) → ∀𝑦𝐵𝑓 ∈ (𝑋𝐻𝑦)(𝑓(⟨𝑋, 𝑋· 𝑦)𝑔) = 𝑓))
1211ss2rabi 3943 . . . . 5 {𝑔 ∈ (𝑋𝐻𝑋) ∣ ∀𝑦𝐵 (∀𝑓 ∈ (𝑦𝐻𝑋)(𝑔(⟨𝑦, 𝑋· 𝑋)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑋𝐻𝑦)(𝑓(⟨𝑋, 𝑋· 𝑦)𝑔) = 𝑓)} ⊆ {𝑔 ∈ (𝑋𝐻𝑋) ∣ ∀𝑦𝐵𝑓 ∈ (𝑋𝐻𝑦)(𝑓(⟨𝑋, 𝑋· 𝑦)𝑔) = 𝑓}
13 catidcl.b . . . . . . 7 𝐵 = (Base‘𝐶)
14 catidcl.h . . . . . . 7 𝐻 = (Hom ‘𝐶)
15 catlid.o . . . . . . 7 · = (comp‘𝐶)
16 catidcl.c . . . . . . 7 (𝜑𝐶 ∈ Cat)
17 catidcl.i . . . . . . 7 1 = (Id‘𝐶)
18 catidcl.x . . . . . . 7 (𝜑𝑋𝐵)
1913, 14, 15, 16, 17, 18cidval 16806 . . . . . 6 (𝜑 → ( 1𝑋) = (𝑔 ∈ (𝑋𝐻𝑋)∀𝑦𝐵 (∀𝑓 ∈ (𝑦𝐻𝑋)(𝑔(⟨𝑦, 𝑋· 𝑋)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑋𝐻𝑦)(𝑓(⟨𝑋, 𝑋· 𝑦)𝑔) = 𝑓)))
2013, 14, 15, 16, 18catideu 16804 . . . . . . 7 (𝜑 → ∃!𝑔 ∈ (𝑋𝐻𝑋)∀𝑦𝐵 (∀𝑓 ∈ (𝑦𝐻𝑋)(𝑔(⟨𝑦, 𝑋· 𝑋)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑋𝐻𝑦)(𝑓(⟨𝑋, 𝑋· 𝑦)𝑔) = 𝑓))
21 riotacl2 6950 . . . . . . 7 (∃!𝑔 ∈ (𝑋𝐻𝑋)∀𝑦𝐵 (∀𝑓 ∈ (𝑦𝐻𝑋)(𝑔(⟨𝑦, 𝑋· 𝑋)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑋𝐻𝑦)(𝑓(⟨𝑋, 𝑋· 𝑦)𝑔) = 𝑓) → (𝑔 ∈ (𝑋𝐻𝑋)∀𝑦𝐵 (∀𝑓 ∈ (𝑦𝐻𝑋)(𝑔(⟨𝑦, 𝑋· 𝑋)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑋𝐻𝑦)(𝑓(⟨𝑋, 𝑋· 𝑦)𝑔) = 𝑓)) ∈ {𝑔 ∈ (𝑋𝐻𝑋) ∣ ∀𝑦𝐵 (∀𝑓 ∈ (𝑦𝐻𝑋)(𝑔(⟨𝑦, 𝑋· 𝑋)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑋𝐻𝑦)(𝑓(⟨𝑋, 𝑋· 𝑦)𝑔) = 𝑓)})
2220, 21syl 17 . . . . . 6 (𝜑 → (𝑔 ∈ (𝑋𝐻𝑋)∀𝑦𝐵 (∀𝑓 ∈ (𝑦𝐻𝑋)(𝑔(⟨𝑦, 𝑋· 𝑋)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑋𝐻𝑦)(𝑓(⟨𝑋, 𝑋· 𝑦)𝑔) = 𝑓)) ∈ {𝑔 ∈ (𝑋𝐻𝑋) ∣ ∀𝑦𝐵 (∀𝑓 ∈ (𝑦𝐻𝑋)(𝑔(⟨𝑦, 𝑋· 𝑋)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑋𝐻𝑦)(𝑓(⟨𝑋, 𝑋· 𝑦)𝑔) = 𝑓)})
2319, 22eqeltrd 2866 . . . . 5 (𝜑 → ( 1𝑋) ∈ {𝑔 ∈ (𝑋𝐻𝑋) ∣ ∀𝑦𝐵 (∀𝑓 ∈ (𝑦𝐻𝑋)(𝑔(⟨𝑦, 𝑋· 𝑋)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑋𝐻𝑦)(𝑓(⟨𝑋, 𝑋· 𝑦)𝑔) = 𝑓)})
2412, 23sseldi 3856 . . . 4 (𝜑 → ( 1𝑋) ∈ {𝑔 ∈ (𝑋𝐻𝑋) ∣ ∀𝑦𝐵𝑓 ∈ (𝑋𝐻𝑦)(𝑓(⟨𝑋, 𝑋· 𝑦)𝑔) = 𝑓})
25 oveq2 6984 . . . . . . . 8 (𝑔 = ( 1𝑋) → (𝑓(⟨𝑋, 𝑋· 𝑦)𝑔) = (𝑓(⟨𝑋, 𝑋· 𝑦)( 1𝑋)))
2625eqeq1d 2780 . . . . . . 7 (𝑔 = ( 1𝑋) → ((𝑓(⟨𝑋, 𝑋· 𝑦)𝑔) = 𝑓 ↔ (𝑓(⟨𝑋, 𝑋· 𝑦)( 1𝑋)) = 𝑓))
27262ralbidv 3149 . . . . . 6 (𝑔 = ( 1𝑋) → (∀𝑦𝐵𝑓 ∈ (𝑋𝐻𝑦)(𝑓(⟨𝑋, 𝑋· 𝑦)𝑔) = 𝑓 ↔ ∀𝑦𝐵𝑓 ∈ (𝑋𝐻𝑦)(𝑓(⟨𝑋, 𝑋· 𝑦)( 1𝑋)) = 𝑓))
2827elrab 3595 . . . . 5 (( 1𝑋) ∈ {𝑔 ∈ (𝑋𝐻𝑋) ∣ ∀𝑦𝐵𝑓 ∈ (𝑋𝐻𝑦)(𝑓(⟨𝑋, 𝑋· 𝑦)𝑔) = 𝑓} ↔ (( 1𝑋) ∈ (𝑋𝐻𝑋) ∧ ∀𝑦𝐵𝑓 ∈ (𝑋𝐻𝑦)(𝑓(⟨𝑋, 𝑋· 𝑦)( 1𝑋)) = 𝑓))
2928simprbi 489 . . . 4 (( 1𝑋) ∈ {𝑔 ∈ (𝑋𝐻𝑋) ∣ ∀𝑦𝐵𝑓 ∈ (𝑋𝐻𝑦)(𝑓(⟨𝑋, 𝑋· 𝑦)𝑔) = 𝑓} → ∀𝑦𝐵𝑓 ∈ (𝑋𝐻𝑦)(𝑓(⟨𝑋, 𝑋· 𝑦)( 1𝑋)) = 𝑓)
3024, 29syl 17 . . 3 (𝜑 → ∀𝑦𝐵𝑓 ∈ (𝑋𝐻𝑦)(𝑓(⟨𝑋, 𝑋· 𝑦)( 1𝑋)) = 𝑓)
31 catlid.y . . 3 (𝜑𝑌𝐵)
328, 30, 31rspcdva 3541 . 2 (𝜑 → ∀𝑓 ∈ (𝑋𝐻𝑌)(𝑓(⟨𝑋, 𝑋· 𝑌)( 1𝑋)) = 𝑓)
33 catlid.f . 2 (𝜑𝐹 ∈ (𝑋𝐻𝑌))
343, 32, 33rspcdva 3541 1 (𝜑 → (𝐹(⟨𝑋, 𝑋· 𝑌)( 1𝑋)) = 𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 387   = wceq 1507  wcel 2050  wral 3088  ∃!wreu 3090  {crab 3092  cop 4447  cfv 6188  crio 6936  (class class class)co 6976  Basecbs 16339  Hom chom 16432  compcco 16433  Catccat 16793  Idccid 16794
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2750  ax-rep 5049  ax-sep 5060  ax-nul 5067  ax-pr 5186
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2759  df-cleq 2771  df-clel 2846  df-nfc 2918  df-ne 2968  df-ral 3093  df-rex 3094  df-reu 3095  df-rmo 3096  df-rab 3097  df-v 3417  df-sbc 3682  df-csb 3787  df-dif 3832  df-un 3834  df-in 3836  df-ss 3843  df-nul 4179  df-if 4351  df-sn 4442  df-pr 4444  df-op 4448  df-uni 4713  df-iun 4794  df-br 4930  df-opab 4992  df-mpt 5009  df-id 5312  df-xp 5413  df-rel 5414  df-cnv 5415  df-co 5416  df-dm 5417  df-rn 5418  df-res 5419  df-ima 5420  df-iota 6152  df-fun 6190  df-fn 6191  df-f 6192  df-f1 6193  df-fo 6194  df-f1o 6195  df-fv 6196  df-riota 6937  df-ov 6979  df-cat 16797  df-cid 16798
This theorem is referenced by:  oppccatid  16847  sectcan  16883  monsect  16911  invisoinvl  16918  rcaninv  16922  subccatid  16974  fucidcl  17093  fucrid  17095  invfuc  17102  arwrid  17191  xpccatid  17296  curf2cl  17339  curfuncf  17346  uncfcurf  17347  hofcl  17367  yonedalem3b  17387
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