Theorem catrid 17736

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	⊢ (𝜑 → 𝐶 ∈ Cat)
catidcl.x	⊢ (𝜑 → 𝑋 ∈ 𝐵)
catlid.o	⊢ · = (comp‘𝐶)
catlid.y	⊢ (𝜑 → 𝑌 ∈ 𝐵)
catlid.f	⊢ (𝜑 → 𝐹 ∈ (𝑋𝐻𝑌))

Assertion

Ref	Expression
catrid	⊢ (𝜑 → (𝐹(⟨𝑋, 𝑋⟩ · 𝑌)( 1 ‘𝑋)) = 𝐹)

Proof of Theorem catrid

Dummy variables 𝑓 𝑔 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		oveq1 7415	. . 3 ⊢ (𝑓 = 𝐹 → (𝑓(⟨𝑋, 𝑋⟩ · 𝑌)( 1 ‘𝑋)) = (𝐹(⟨𝑋, 𝑋⟩ · 𝑌)( 1 ‘𝑋)))
2		id 23	. . 3 ⊢ (𝑓 = 𝐹 → 𝑓 = 𝐹)
3	1, 2	eqeq12d 2785	. 2 ⊢ (𝑓 = 𝐹 → ((𝑓(⟨𝑋, 𝑋⟩ · 𝑌)( 1 ‘𝑋)) = 𝑓 ↔ (𝐹(⟨𝑋, 𝑋⟩ · 𝑌)( 1 ‘𝑋)) = 𝐹))
4		oveq2 7416	. . . 4 ⊢ (𝑦 = 𝑌 → (𝑋𝐻𝑦) = (𝑋𝐻𝑌))
5		oveq2 7416	. . . . . 6 ⊢ (𝑦 = 𝑌 → (⟨𝑋, 𝑋⟩ · 𝑦) = (⟨𝑋, 𝑋⟩ · 𝑌))
6	5	oveqd 7425	. . . . 5 ⊢ (𝑦 = 𝑌 → (𝑓(⟨𝑋, 𝑋⟩ · 𝑦)( 1 ‘𝑋)) = (𝑓(⟨𝑋, 𝑋⟩ · 𝑌)( 1 ‘𝑋)))
7	6	eqeq1d 2771	. . . 4 ⊢ (𝑦 = 𝑌 → ((𝑓(⟨𝑋, 𝑋⟩ · 𝑦)( 1 ‘𝑋)) = 𝑓 ↔ (𝑓(⟨𝑋, 𝑋⟩ · 𝑌)( 1 ‘𝑋)) = 𝑓))
8	4, 7	raleqbidv 3345	. . 3 ⊢ (𝑦 = 𝑌 → (∀𝑓 ∈ (𝑋𝐻𝑦)(𝑓(⟨𝑋, 𝑋⟩ · 𝑦)( 1 ‘𝑋)) = 𝑓 ↔ ∀𝑓 ∈ (𝑋𝐻𝑌)(𝑓(⟨𝑋, 𝑋⟩ · 𝑌)( 1 ‘𝑋)) = 𝑓))
9		simpr 489	. . . . . . . 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)
10	9	ralimi 3108	. . . . . . 7 ⊢ (∀𝑦 ∈ 𝐵 (∀𝑓 ∈ (𝑦𝐻𝑋)(𝑔(⟨𝑦, 𝑋⟩ · 𝑋)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑋𝐻𝑦)(𝑓(⟨𝑋, 𝑋⟩ · 𝑦)𝑔) = 𝑓) → ∀𝑦 ∈ 𝐵 ∀𝑓 ∈ (𝑋𝐻𝑦)(𝑓(⟨𝑋, 𝑋⟩ · 𝑦)𝑔) = 𝑓)
11	10	a1i 11	. . . . . 6 ⊢ (𝑔 ∈ (𝑋𝐻𝑋) → (∀𝑦 ∈ 𝐵 (∀𝑓 ∈ (𝑦𝐻𝑋)(𝑔(⟨𝑦, 𝑋⟩ · 𝑋)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑋𝐻𝑦)(𝑓(⟨𝑋, 𝑋⟩ · 𝑦)𝑔) = 𝑓) → ∀𝑦 ∈ 𝐵 ∀𝑓 ∈ (𝑋𝐻𝑦)(𝑓(⟨𝑋, 𝑋⟩ · 𝑦)𝑔) = 𝑓))
12	11	ss2rabi 4038	. . . . 5 ⊢ {𝑔 ∈ (𝑋𝐻𝑋) ∣ ∀𝑦 ∈ 𝐵 (∀𝑓 ∈ (𝑦𝐻𝑋)(𝑔(⟨𝑦, 𝑋⟩ · 𝑋)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑋𝐻𝑦)(𝑓(⟨𝑋, 𝑋⟩ · 𝑦)𝑔) = 𝑓)} ⊆ {𝑔 ∈ (𝑋𝐻𝑋) ∣ ∀𝑦 ∈ 𝐵 ∀𝑓 ∈ (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 ⊢ (𝜑 → 𝐶 ∈ Cat)
17		catidcl.i	. . . . . . 7 ⊢ 1 = (Id‘𝐶)
18		catidcl.x	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
19	13, 14, 15, 16, 17, 18	cidval 17729	. . . . . 6 ⊢ (𝜑 → ( 1 ‘𝑋) = (℩𝑔 ∈ (𝑋𝐻𝑋)∀𝑦 ∈ 𝐵 (∀𝑓 ∈ (𝑦𝐻𝑋)(𝑔(⟨𝑦, 𝑋⟩ · 𝑋)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑋𝐻𝑦)(𝑓(⟨𝑋, 𝑋⟩ · 𝑦)𝑔) = 𝑓)))
20	13, 14, 15, 16, 18	catideu 17727	. . . . . . 7 ⊢ (𝜑 → ∃!𝑔 ∈ (𝑋𝐻𝑋)∀𝑦 ∈ 𝐵 (∀𝑓 ∈ (𝑦𝐻𝑋)(𝑔(⟨𝑦, 𝑋⟩ · 𝑋)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑋𝐻𝑦)(𝑓(⟨𝑋, 𝑋⟩ · 𝑦)𝑔) = 𝑓))
21		riotacl2 7381	. . . . . . 7 ⊢ (∃!𝑔 ∈ (𝑋𝐻𝑋)∀𝑦 ∈ 𝐵 (∀𝑓 ∈ (𝑦𝐻𝑋)(𝑔(⟨𝑦, 𝑋⟩ · 𝑋)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑋𝐻𝑦)(𝑓(⟨𝑋, 𝑋⟩ · 𝑦)𝑔) = 𝑓) → (℩𝑔 ∈ (𝑋𝐻𝑋)∀𝑦 ∈ 𝐵 (∀𝑓 ∈ (𝑦𝐻𝑋)(𝑔(⟨𝑦, 𝑋⟩ · 𝑋)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑋𝐻𝑦)(𝑓(⟨𝑋, 𝑋⟩ · 𝑦)𝑔) = 𝑓)) ∈ {𝑔 ∈ (𝑋𝐻𝑋) ∣ ∀𝑦 ∈ 𝐵 (∀𝑓 ∈ (𝑦𝐻𝑋)(𝑔(⟨𝑦, 𝑋⟩ · 𝑋)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑋𝐻𝑦)(𝑓(⟨𝑋, 𝑋⟩ · 𝑦)𝑔) = 𝑓)})
22	20, 21	syl 18	. . . . . 6 ⊢ (𝜑 → (℩𝑔 ∈ (𝑋𝐻𝑋)∀𝑦 ∈ 𝐵 (∀𝑓 ∈ (𝑦𝐻𝑋)(𝑔(⟨𝑦, 𝑋⟩ · 𝑋)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑋𝐻𝑦)(𝑓(⟨𝑋, 𝑋⟩ · 𝑦)𝑔) = 𝑓)) ∈ {𝑔 ∈ (𝑋𝐻𝑋) ∣ ∀𝑦 ∈ 𝐵 (∀𝑓 ∈ (𝑦𝐻𝑋)(𝑔(⟨𝑦, 𝑋⟩ · 𝑋)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑋𝐻𝑦)(𝑓(⟨𝑋, 𝑋⟩ · 𝑦)𝑔) = 𝑓)})
23	19, 22	eqeltrd 2869	. . . . 5 ⊢ (𝜑 → ( 1 ‘𝑋) ∈ {𝑔 ∈ (𝑋𝐻𝑋) ∣ ∀𝑦 ∈ 𝐵 (∀𝑓 ∈ (𝑦𝐻𝑋)(𝑔(⟨𝑦, 𝑋⟩ · 𝑋)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑋𝐻𝑦)(𝑓(⟨𝑋, 𝑋⟩ · 𝑦)𝑔) = 𝑓)})
24	12, 23	sselid 3943	. . . 4 ⊢ (𝜑 → ( 1 ‘𝑋) ∈ {𝑔 ∈ (𝑋𝐻𝑋) ∣ ∀𝑦 ∈ 𝐵 ∀𝑓 ∈ (𝑋𝐻𝑦)(𝑓(⟨𝑋, 𝑋⟩ · 𝑦)𝑔) = 𝑓})
25		oveq2 7416	. . . . . . . 8 ⊢ (𝑔 = ( 1 ‘𝑋) → (𝑓(⟨𝑋, 𝑋⟩ · 𝑦)𝑔) = (𝑓(⟨𝑋, 𝑋⟩ · 𝑦)( 1 ‘𝑋)))
26	25	eqeq1d 2771	. . . . . . 7 ⊢ (𝑔 = ( 1 ‘𝑋) → ((𝑓(⟨𝑋, 𝑋⟩ · 𝑦)𝑔) = 𝑓 ↔ (𝑓(⟨𝑋, 𝑋⟩ · 𝑦)( 1 ‘𝑋)) = 𝑓))
27	26	2ralbidv 3235	. . . . . 6 ⊢ (𝑔 = ( 1 ‘𝑋) → (∀𝑦 ∈ 𝐵 ∀𝑓 ∈ (𝑋𝐻𝑦)(𝑓(⟨𝑋, 𝑋⟩ · 𝑦)𝑔) = 𝑓 ↔ ∀𝑦 ∈ 𝐵 ∀𝑓 ∈ (𝑋𝐻𝑦)(𝑓(⟨𝑋, 𝑋⟩ · 𝑦)( 1 ‘𝑋)) = 𝑓))
28	27	elrab 3659	. . . . 5 ⊢ (( 1 ‘𝑋) ∈ {𝑔 ∈ (𝑋𝐻𝑋) ∣ ∀𝑦 ∈ 𝐵 ∀𝑓 ∈ (𝑋𝐻𝑦)(𝑓(⟨𝑋, 𝑋⟩ · 𝑦)𝑔) = 𝑓} ↔ (( 1 ‘𝑋) ∈ (𝑋𝐻𝑋) ∧ ∀𝑦 ∈ 𝐵 ∀𝑓 ∈ (𝑋𝐻𝑦)(𝑓(⟨𝑋, 𝑋⟩ · 𝑦)( 1 ‘𝑋)) = 𝑓))
29	28	simprbi 502	. . . 4 ⊢ (( 1 ‘𝑋) ∈ {𝑔 ∈ (𝑋𝐻𝑋) ∣ ∀𝑦 ∈ 𝐵 ∀𝑓 ∈ (𝑋𝐻𝑦)(𝑓(⟨𝑋, 𝑋⟩ · 𝑦)𝑔) = 𝑓} → ∀𝑦 ∈ 𝐵 ∀𝑓 ∈ (𝑋𝐻𝑦)(𝑓(⟨𝑋, 𝑋⟩ · 𝑦)( 1 ‘𝑋)) = 𝑓)
30	24, 29	syl 18	. . 3 ⊢ (𝜑 → ∀𝑦 ∈ 𝐵 ∀𝑓 ∈ (𝑋𝐻𝑦)(𝑓(⟨𝑋, 𝑋⟩ · 𝑦)( 1 ‘𝑋)) = 𝑓)
31		catlid.y	. . 3 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
32	8, 30, 31	rspcdva 3591	. 2 ⊢ (𝜑 → ∀𝑓 ∈ (𝑋𝐻𝑌)(𝑓(⟨𝑋, 𝑋⟩ · 𝑌)( 1 ‘𝑋)) = 𝑓)
33		catlid.f	. 2 ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐻𝑌))
34	3, 32, 33	rspcdva 3591	1 ⊢ (𝜑 → (𝐹(⟨𝑋, 𝑋⟩ · 𝑌)( 1 ‘𝑋)) = 𝐹)

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1567   ∈ wcel 2149  ∀wral 3085  ∃!wreu 3374  {crab 3423  ⟨cop 4597  ‘cfv 6533  ℩crio 7364  (class class class)co 7408  Basecbs 17265  Hom chom 17317  compcco 17318  Catccat 17716  Idccid 17717

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5239  ax-sep 5258  ax-nul 5268  ax-pr 5402

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-rmo 3376  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-iun 4959  df-br 5111  df-opab 5175  df-mpt 5194  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-riota 7365  df-ov 7411  df-cat 17720  df-cid 17721

This theorem is referenced by:  oppccatid  17771  sectcan  17808  monsect  17836  invisoinvl  17843  rcaninv  17847  subccatid  17899  fucidcl  18021  fucrid  18023  invfuc  18030  arwrid  18126  xpccatid  18240  curf2cl  18283  curfuncf  18290  uncfcurf  18291  hofcl  18311  yonedalem3b  18331  bj-endmnd  37845  endmndlem  49671  idepi  49677  upeu2lem  49684  fucorid  50018  precofvalALT  50024  concom  50319