Theorem catrid 17864

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 7428	. . 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 2776	. 2 ⊢ (𝑓 = 𝐹 → ((𝑓(⟨𝑋, 𝑋⟩ · 𝑌)( 1 ‘𝑋)) = 𝑓 ↔ (𝐹(⟨𝑋, 𝑋⟩ · 𝑌)( 1 ‘𝑋)) = 𝐹))
4		oveq2 7429	. . . 4 ⊢ (𝑦 = 𝑌 → (𝑋𝐻𝑦) = (𝑋𝐻𝑌))
5		oveq2 7429	. . . . . 6 ⊢ (𝑦 = 𝑌 → (⟨𝑋, 𝑋⟩ · 𝑦) = (⟨𝑋, 𝑋⟩ · 𝑌))
6	5	oveqd 7438	. . . . 5 ⊢ (𝑦 = 𝑌 → (𝑓(⟨𝑋, 𝑋⟩ · 𝑦)( 1 ‘𝑋)) = (𝑓(⟨𝑋, 𝑋⟩ · 𝑌)( 1 ‘𝑋)))
7	6	eqeq1d 2762	. . . 4 ⊢ (𝑦 = 𝑌 → ((𝑓(⟨𝑋, 𝑋⟩ · 𝑦)( 1 ‘𝑋)) = 𝑓 ↔ (𝑓(⟨𝑋, 𝑋⟩ · 𝑌)( 1 ‘𝑋)) = 𝑓))
8	4, 7	raleqbidv 3334	. . 3 ⊢ (𝑦 = 𝑌 → (∀𝑓 ∈ (𝑋𝐻𝑦)(𝑓(⟨𝑋, 𝑋⟩ · 𝑦)( 1 ‘𝑋)) = 𝑓 ↔ ∀𝑓 ∈ (𝑋𝐻𝑌)(𝑓(⟨𝑋, 𝑋⟩ · 𝑌)( 1 ‘𝑋)) = 𝑓))
9		simpr 490	. . . . . . . 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 3099	. . . . . . 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 4001	. . . . 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 17857	. . . . . 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 17855	. . . . . . 7 ⊢ (𝜑 → ∃!𝑔 ∈ (𝑋𝐻𝑋)∀𝑦 ∈ 𝐵 (∀𝑓 ∈ (𝑦𝐻𝑋)(𝑔(⟨𝑦, 𝑋⟩ · 𝑋)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑋𝐻𝑦)(𝑓(⟨𝑋, 𝑋⟩ · 𝑦)𝑔) = 𝑓))
21		riotacl2 7394	. . . . . . 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 2860	. . . . 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 3906	. . . 4 ⊢ (𝜑 → ( 1 ‘𝑋) ∈ {𝑔 ∈ (𝑋𝐻𝑋) ∣ ∀𝑦 ∈ 𝐵 ∀𝑓 ∈ (𝑋𝐻𝑦)(𝑓(⟨𝑋, 𝑋⟩ · 𝑦)𝑔) = 𝑓})
25		oveq2 7429	. . . . . . . 8 ⊢ (𝑔 = ( 1 ‘𝑋) → (𝑓(⟨𝑋, 𝑋⟩ · 𝑦)𝑔) = (𝑓(⟨𝑋, 𝑋⟩ · 𝑦)( 1 ‘𝑋)))
26	25	eqeq1d 2762	. . . . . . 7 ⊢ (𝑔 = ( 1 ‘𝑋) → ((𝑓(⟨𝑋, 𝑋⟩ · 𝑦)𝑔) = 𝑓 ↔ (𝑓(⟨𝑋, 𝑋⟩ · 𝑦)( 1 ‘𝑋)) = 𝑓))
27	26	2ralbidv 3226	. . . . . 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 3644	. . . . 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 503	. . . 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 3577	. 2 ⊢ (𝜑 → ∀𝑓 ∈ (𝑋𝐻𝑌)(𝑓(⟨𝑋, 𝑋⟩ · 𝑌)( 1 ‘𝑋)) = 𝑓)
33		catlid.f	. 2 ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐻𝑌))
34	3, 32, 33	rspcdva 3577	1 ⊢ (𝜑 → (𝐹(⟨𝑋, 𝑋⟩ · 𝑌)( 1 ‘𝑋)) = 𝐹)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1569   ∈ wcel 2144  ∀wral 3076  ∃!wreu 3363  {crab 3412  ⟨cop 4834  ‘cfv 6539  ℩crio 7377  (class class class)co 7421  Basecbs 17393  Hom chom 17445  compcco 17446  Catccat 17844  Idccid 17845

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-iota 6494  df-fun 6541  df-fn 6542  df-f 6543  df-f1 6544  df-fo 6545  df-f1o 6546  df-fv 6547  df-riota 7378  df-ov 7424  df-cat 17848  df-cid 17849

This theorem is used by:  oppccatid  17899  sectcan  17936  monsect  17964  invisoinvl  17971  rcaninv  17975  subccatid  18027  fucidcl  18149  fucrid  18151  invfuc  18158  arwrid  18254  xpccatid  18368  curf2cl  18411  curfuncf  18418  uncfcurf  18419  hofcl  18439  yonedalem3b  18459  bj-endmnd  38333  endmndlem  50223  idepi  50229  upeu2lem  50236  fucorid  50578  precofvalALT  50597  parcoval2  50860  concom  50940