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Theorem catrid 17491
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 7345 . . 3 (𝑓 = 𝐹 → (𝑓(⟨𝑋, 𝑋· 𝑌)( 1𝑋)) = (𝐹(⟨𝑋, 𝑋· 𝑌)( 1𝑋)))
2 id 22 . . 3 (𝑓 = 𝐹𝑓 = 𝐹)
31, 2eqeq12d 2752 . 2 (𝑓 = 𝐹 → ((𝑓(⟨𝑋, 𝑋· 𝑌)( 1𝑋)) = 𝑓 ↔ (𝐹(⟨𝑋, 𝑋· 𝑌)( 1𝑋)) = 𝐹))
4 oveq2 7346 . . . 4 (𝑦 = 𝑌 → (𝑋𝐻𝑦) = (𝑋𝐻𝑌))
5 oveq2 7346 . . . . . 6 (𝑦 = 𝑌 → (⟨𝑋, 𝑋· 𝑦) = (⟨𝑋, 𝑋· 𝑌))
65oveqd 7355 . . . . 5 (𝑦 = 𝑌 → (𝑓(⟨𝑋, 𝑋· 𝑦)( 1𝑋)) = (𝑓(⟨𝑋, 𝑋· 𝑌)( 1𝑋)))
76eqeq1d 2738 . . . 4 (𝑦 = 𝑌 → ((𝑓(⟨𝑋, 𝑋· 𝑦)( 1𝑋)) = 𝑓 ↔ (𝑓(⟨𝑋, 𝑋· 𝑌)( 1𝑋)) = 𝑓))
84, 7raleqbidv 3315 . . 3 (𝑦 = 𝑌 → (∀𝑓 ∈ (𝑋𝐻𝑦)(𝑓(⟨𝑋, 𝑋· 𝑦)( 1𝑋)) = 𝑓 ↔ ∀𝑓 ∈ (𝑋𝐻𝑌)(𝑓(⟨𝑋, 𝑋· 𝑌)( 1𝑋)) = 𝑓))
9 simpr 485 . . . . . . . 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 3082 . . . . . . 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 4022 . . . . 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 17484 . . . . . 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 17482 . . . . . . 7 (𝜑 → ∃!𝑔 ∈ (𝑋𝐻𝑋)∀𝑦𝐵 (∀𝑓 ∈ (𝑦𝐻𝑋)(𝑔(⟨𝑦, 𝑋· 𝑋)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑋𝐻𝑦)(𝑓(⟨𝑋, 𝑋· 𝑦)𝑔) = 𝑓))
21 riotacl2 7311 . . . . . . 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 2837 . . . . 5 (𝜑 → ( 1𝑋) ∈ {𝑔 ∈ (𝑋𝐻𝑋) ∣ ∀𝑦𝐵 (∀𝑓 ∈ (𝑦𝐻𝑋)(𝑔(⟨𝑦, 𝑋· 𝑋)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑋𝐻𝑦)(𝑓(⟨𝑋, 𝑋· 𝑦)𝑔) = 𝑓)})
2412, 23sselid 3930 . . . 4 (𝜑 → ( 1𝑋) ∈ {𝑔 ∈ (𝑋𝐻𝑋) ∣ ∀𝑦𝐵𝑓 ∈ (𝑋𝐻𝑦)(𝑓(⟨𝑋, 𝑋· 𝑦)𝑔) = 𝑓})
25 oveq2 7346 . . . . . . . 8 (𝑔 = ( 1𝑋) → (𝑓(⟨𝑋, 𝑋· 𝑦)𝑔) = (𝑓(⟨𝑋, 𝑋· 𝑦)( 1𝑋)))
2625eqeq1d 2738 . . . . . . 7 (𝑔 = ( 1𝑋) → ((𝑓(⟨𝑋, 𝑋· 𝑦)𝑔) = 𝑓 ↔ (𝑓(⟨𝑋, 𝑋· 𝑦)( 1𝑋)) = 𝑓))
27262ralbidv 3208 . . . . . 6 (𝑔 = ( 1𝑋) → (∀𝑦𝐵𝑓 ∈ (𝑋𝐻𝑦)(𝑓(⟨𝑋, 𝑋· 𝑦)𝑔) = 𝑓 ↔ ∀𝑦𝐵𝑓 ∈ (𝑋𝐻𝑦)(𝑓(⟨𝑋, 𝑋· 𝑦)( 1𝑋)) = 𝑓))
2827elrab 3634 . . . . 5 (( 1𝑋) ∈ {𝑔 ∈ (𝑋𝐻𝑋) ∣ ∀𝑦𝐵𝑓 ∈ (𝑋𝐻𝑦)(𝑓(⟨𝑋, 𝑋· 𝑦)𝑔) = 𝑓} ↔ (( 1𝑋) ∈ (𝑋𝐻𝑋) ∧ ∀𝑦𝐵𝑓 ∈ (𝑋𝐻𝑦)(𝑓(⟨𝑋, 𝑋· 𝑦)( 1𝑋)) = 𝑓))
2928simprbi 497 . . . 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 3571 . 2 (𝜑 → ∀𝑓 ∈ (𝑋𝐻𝑌)(𝑓(⟨𝑋, 𝑋· 𝑌)( 1𝑋)) = 𝑓)
33 catlid.f . 2 (𝜑𝐹 ∈ (𝑋𝐻𝑌))
343, 32, 33rspcdva 3571 1 (𝜑 → (𝐹(⟨𝑋, 𝑋· 𝑌)( 1𝑋)) = 𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1540  wcel 2105  wral 3061  ∃!wreu 3347  {crab 3403  cop 4580  cfv 6480  crio 7293  (class class class)co 7338  Basecbs 17010  Hom chom 17071  compcco 17072  Catccat 17471  Idccid 17472
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 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2707  ax-rep 5230  ax-sep 5244  ax-nul 5251  ax-pr 5373
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3349  df-reu 3350  df-rab 3404  df-v 3443  df-sbc 3728  df-csb 3844  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4271  df-if 4475  df-sn 4575  df-pr 4577  df-op 4581  df-uni 4854  df-iun 4944  df-br 5094  df-opab 5156  df-mpt 5177  df-id 5519  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-iota 6432  df-fun 6482  df-fn 6483  df-f 6484  df-f1 6485  df-fo 6486  df-f1o 6487  df-fv 6488  df-riota 7294  df-ov 7341  df-cat 17475  df-cid 17476
This theorem is referenced by:  oppccatid  17528  sectcan  17565  monsect  17593  invisoinvl  17600  rcaninv  17604  subccatid  17659  fucidcl  17781  fucrid  17783  invfuc  17790  arwrid  17886  xpccatid  18003  curf2cl  18047  curfuncf  18054  uncfcurf  18055  hofcl  18075  yonedalem3b  18095  bj-endmnd  35645  endmndlem  46714  idepi  46716
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