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Theorem catrid 17564
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 7364 . . 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 7365 . . . 4 (𝑦 = 𝑌 → (𝑋𝐻𝑦) = (𝑋𝐻𝑌))
5 oveq2 7365 . . . . . 6 (𝑦 = 𝑌 → (⟨𝑋, 𝑋· 𝑦) = (⟨𝑋, 𝑋· 𝑌))
65oveqd 7374 . . . . 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 3319 . . 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 3086 . . . . . . 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 4034 . . . . 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 17557 . . . . . 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 17555 . . . . . . 7 (𝜑 → ∃!𝑔 ∈ (𝑋𝐻𝑋)∀𝑦𝐵 (∀𝑓 ∈ (𝑦𝐻𝑋)(𝑔(⟨𝑦, 𝑋· 𝑋)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑋𝐻𝑦)(𝑓(⟨𝑋, 𝑋· 𝑦)𝑔) = 𝑓))
21 riotacl2 7330 . . . . . . 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 2838 . . . . 5 (𝜑 → ( 1𝑋) ∈ {𝑔 ∈ (𝑋𝐻𝑋) ∣ ∀𝑦𝐵 (∀𝑓 ∈ (𝑦𝐻𝑋)(𝑔(⟨𝑦, 𝑋· 𝑋)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑋𝐻𝑦)(𝑓(⟨𝑋, 𝑋· 𝑦)𝑔) = 𝑓)})
2412, 23sselid 3942 . . . 4 (𝜑 → ( 1𝑋) ∈ {𝑔 ∈ (𝑋𝐻𝑋) ∣ ∀𝑦𝐵𝑓 ∈ (𝑋𝐻𝑦)(𝑓(⟨𝑋, 𝑋· 𝑦)𝑔) = 𝑓})
25 oveq2 7365 . . . . . . . 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 3212 . . . . . 6 (𝑔 = ( 1𝑋) → (∀𝑦𝐵𝑓 ∈ (𝑋𝐻𝑦)(𝑓(⟨𝑋, 𝑋· 𝑦)𝑔) = 𝑓 ↔ ∀𝑦𝐵𝑓 ∈ (𝑋𝐻𝑦)(𝑓(⟨𝑋, 𝑋· 𝑦)( 1𝑋)) = 𝑓))
2827elrab 3645 . . . . 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 3582 . 2 (𝜑 → ∀𝑓 ∈ (𝑋𝐻𝑌)(𝑓(⟨𝑋, 𝑋· 𝑌)( 1𝑋)) = 𝑓)
33 catlid.f . 2 (𝜑𝐹 ∈ (𝑋𝐻𝑌))
343, 32, 33rspcdva 3582 1 (𝜑 → (𝐹(⟨𝑋, 𝑋· 𝑌)( 1𝑋)) = 𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1541  wcel 2106  wral 3064  ∃!wreu 3351  {crab 3407  cop 4592  cfv 6496  crio 7312  (class class class)co 7357  Basecbs 17083  Hom chom 17144  compcco 17145  Catccat 17544  Idccid 17545
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pr 5384
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-cat 17548  df-cid 17549
This theorem is referenced by:  oppccatid  17601  sectcan  17638  monsect  17666  invisoinvl  17673  rcaninv  17677  subccatid  17732  fucidcl  17854  fucrid  17856  invfuc  17863  arwrid  17959  xpccatid  18076  curf2cl  18120  curfuncf  18127  uncfcurf  18128  hofcl  18148  yonedalem3b  18168  bj-endmnd  35789  endmndlem  47025  idepi  47027
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