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Theorem catrid 16951
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 7146 . . 3 (𝑓 = 𝐹 → (𝑓(⟨𝑋, 𝑋· 𝑌)( 1𝑋)) = (𝐹(⟨𝑋, 𝑋· 𝑌)( 1𝑋)))
2 id 22 . . 3 (𝑓 = 𝐹𝑓 = 𝐹)
31, 2eqeq12d 2817 . 2 (𝑓 = 𝐹 → ((𝑓(⟨𝑋, 𝑋· 𝑌)( 1𝑋)) = 𝑓 ↔ (𝐹(⟨𝑋, 𝑋· 𝑌)( 1𝑋)) = 𝐹))
4 oveq2 7147 . . . 4 (𝑦 = 𝑌 → (𝑋𝐻𝑦) = (𝑋𝐻𝑌))
5 oveq2 7147 . . . . . 6 (𝑦 = 𝑌 → (⟨𝑋, 𝑋· 𝑦) = (⟨𝑋, 𝑋· 𝑌))
65oveqd 7156 . . . . 5 (𝑦 = 𝑌 → (𝑓(⟨𝑋, 𝑋· 𝑦)( 1𝑋)) = (𝑓(⟨𝑋, 𝑋· 𝑌)( 1𝑋)))
76eqeq1d 2803 . . . 4 (𝑦 = 𝑌 → ((𝑓(⟨𝑋, 𝑋· 𝑦)( 1𝑋)) = 𝑓 ↔ (𝑓(⟨𝑋, 𝑋· 𝑌)( 1𝑋)) = 𝑓))
84, 7raleqbidv 3357 . . 3 (𝑦 = 𝑌 → (∀𝑓 ∈ (𝑋𝐻𝑦)(𝑓(⟨𝑋, 𝑋· 𝑦)( 1𝑋)) = 𝑓 ↔ ∀𝑓 ∈ (𝑋𝐻𝑌)(𝑓(⟨𝑋, 𝑋· 𝑌)( 1𝑋)) = 𝑓))
9 simpr 488 . . . . . . . 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 3131 . . . . . . 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 4007 . . . . 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 16944 . . . . . 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 16942 . . . . . . 7 (𝜑 → ∃!𝑔 ∈ (𝑋𝐻𝑋)∀𝑦𝐵 (∀𝑓 ∈ (𝑦𝐻𝑋)(𝑔(⟨𝑦, 𝑋· 𝑋)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑋𝐻𝑦)(𝑓(⟨𝑋, 𝑋· 𝑦)𝑔) = 𝑓))
21 riotacl2 7113 . . . . . . 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 2893 . . . . 5 (𝜑 → ( 1𝑋) ∈ {𝑔 ∈ (𝑋𝐻𝑋) ∣ ∀𝑦𝐵 (∀𝑓 ∈ (𝑦𝐻𝑋)(𝑔(⟨𝑦, 𝑋· 𝑋)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑋𝐻𝑦)(𝑓(⟨𝑋, 𝑋· 𝑦)𝑔) = 𝑓)})
2412, 23sseldi 3916 . . . 4 (𝜑 → ( 1𝑋) ∈ {𝑔 ∈ (𝑋𝐻𝑋) ∣ ∀𝑦𝐵𝑓 ∈ (𝑋𝐻𝑦)(𝑓(⟨𝑋, 𝑋· 𝑦)𝑔) = 𝑓})
25 oveq2 7147 . . . . . . . 8 (𝑔 = ( 1𝑋) → (𝑓(⟨𝑋, 𝑋· 𝑦)𝑔) = (𝑓(⟨𝑋, 𝑋· 𝑦)( 1𝑋)))
2625eqeq1d 2803 . . . . . . 7 (𝑔 = ( 1𝑋) → ((𝑓(⟨𝑋, 𝑋· 𝑦)𝑔) = 𝑓 ↔ (𝑓(⟨𝑋, 𝑋· 𝑦)( 1𝑋)) = 𝑓))
27262ralbidv 3167 . . . . . 6 (𝑔 = ( 1𝑋) → (∀𝑦𝐵𝑓 ∈ (𝑋𝐻𝑦)(𝑓(⟨𝑋, 𝑋· 𝑦)𝑔) = 𝑓 ↔ ∀𝑦𝐵𝑓 ∈ (𝑋𝐻𝑦)(𝑓(⟨𝑋, 𝑋· 𝑦)( 1𝑋)) = 𝑓))
2827elrab 3631 . . . . 5 (( 1𝑋) ∈ {𝑔 ∈ (𝑋𝐻𝑋) ∣ ∀𝑦𝐵𝑓 ∈ (𝑋𝐻𝑦)(𝑓(⟨𝑋, 𝑋· 𝑦)𝑔) = 𝑓} ↔ (( 1𝑋) ∈ (𝑋𝐻𝑋) ∧ ∀𝑦𝐵𝑓 ∈ (𝑋𝐻𝑦)(𝑓(⟨𝑋, 𝑋· 𝑦)( 1𝑋)) = 𝑓))
2928simprbi 500 . . . 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 3576 . 2 (𝜑 → ∀𝑓 ∈ (𝑋𝐻𝑌)(𝑓(⟨𝑋, 𝑋· 𝑌)( 1𝑋)) = 𝑓)
33 catlid.f . 2 (𝜑𝐹 ∈ (𝑋𝐻𝑌))
343, 32, 33rspcdva 3576 1 (𝜑 → (𝐹(⟨𝑋, 𝑋· 𝑌)( 1𝑋)) = 𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2112  wral 3109  ∃!wreu 3111  {crab 3113  cop 4534  cfv 6328  crio 7096  (class class class)co 7139  Basecbs 16479  Hom chom 16572  compcco 16573  Catccat 16931  Idccid 16932
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-pr 5298
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-sn 4529  df-pr 4531  df-op 4535  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-cat 16935  df-cid 16936
This theorem is referenced by:  oppccatid  16985  sectcan  17021  monsect  17049  invisoinvl  17056  rcaninv  17060  subccatid  17112  fucidcl  17231  fucrid  17233  invfuc  17240  arwrid  17329  xpccatid  17434  curf2cl  17477  curfuncf  17484  uncfcurf  17485  hofcl  17505  yonedalem3b  17525  bj-endmnd  34733
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