Theorem catrid 17641

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 7367	. . 3 ⊢ (𝑓 = 𝐹 → (𝑓(⟨𝑋, 𝑋⟩ · 𝑌)( 1 ‘𝑋)) = (𝐹(⟨𝑋, 𝑋⟩ · 𝑌)( 1 ‘𝑋)))
2		id 22	. . 3 ⊢ (𝑓 = 𝐹 → 𝑓 = 𝐹)
3	1, 2	eqeq12d 2753	. 2 ⊢ (𝑓 = 𝐹 → ((𝑓(⟨𝑋, 𝑋⟩ · 𝑌)( 1 ‘𝑋)) = 𝑓 ↔ (𝐹(⟨𝑋, 𝑋⟩ · 𝑌)( 1 ‘𝑋)) = 𝐹))
4		oveq2 7368	. . . 4 ⊢ (𝑦 = 𝑌 → (𝑋𝐻𝑦) = (𝑋𝐻𝑌))
5		oveq2 7368	. . . . . 6 ⊢ (𝑦 = 𝑌 → (⟨𝑋, 𝑋⟩ · 𝑦) = (⟨𝑋, 𝑋⟩ · 𝑌))
6	5	oveqd 7377	. . . . 5 ⊢ (𝑦 = 𝑌 → (𝑓(⟨𝑋, 𝑋⟩ · 𝑦)( 1 ‘𝑋)) = (𝑓(⟨𝑋, 𝑋⟩ · 𝑌)( 1 ‘𝑋)))
7	6	eqeq1d 2739	. . . 4 ⊢ (𝑦 = 𝑌 → ((𝑓(⟨𝑋, 𝑋⟩ · 𝑦)( 1 ‘𝑋)) = 𝑓 ↔ (𝑓(⟨𝑋, 𝑋⟩ · 𝑌)( 1 ‘𝑋)) = 𝑓))
8	4, 7	raleqbidv 3312	. . 3 ⊢ (𝑦 = 𝑌 → (∀𝑓 ∈ (𝑋𝐻𝑦)(𝑓(⟨𝑋, 𝑋⟩ · 𝑦)( 1 ‘𝑋)) = 𝑓 ↔ ∀𝑓 ∈ (𝑋𝐻𝑌)(𝑓(⟨𝑋, 𝑋⟩ · 𝑌)( 1 ‘𝑋)) = 𝑓))
9		simpr 484	. . . . . . . 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 3075	. . . . . . 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 4017	. . . . 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 17634	. . . . . 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 17632	. . . . . . 7 ⊢ (𝜑 → ∃!𝑔 ∈ (𝑋𝐻𝑋)∀𝑦 ∈ 𝐵 (∀𝑓 ∈ (𝑦𝐻𝑋)(𝑔(⟨𝑦, 𝑋⟩ · 𝑋)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑋𝐻𝑦)(𝑓(⟨𝑋, 𝑋⟩ · 𝑦)𝑔) = 𝑓))
21		riotacl2 7333	. . . . . . 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 17	. . . . . 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 2837	. . . . 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 3920	. . . 4 ⊢ (𝜑 → ( 1 ‘𝑋) ∈ {𝑔 ∈ (𝑋𝐻𝑋) ∣ ∀𝑦 ∈ 𝐵 ∀𝑓 ∈ (𝑋𝐻𝑦)(𝑓(⟨𝑋, 𝑋⟩ · 𝑦)𝑔) = 𝑓})
25		oveq2 7368	. . . . . . . 8 ⊢ (𝑔 = ( 1 ‘𝑋) → (𝑓(⟨𝑋, 𝑋⟩ · 𝑦)𝑔) = (𝑓(⟨𝑋, 𝑋⟩ · 𝑦)( 1 ‘𝑋)))
26	25	eqeq1d 2739	. . . . . . 7 ⊢ (𝑔 = ( 1 ‘𝑋) → ((𝑓(⟨𝑋, 𝑋⟩ · 𝑦)𝑔) = 𝑓 ↔ (𝑓(⟨𝑋, 𝑋⟩ · 𝑦)( 1 ‘𝑋)) = 𝑓))
27	26	2ralbidv 3202	. . . . . 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 3635	. . . . 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 497	. . . 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 17	. . 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 3566	. 2 ⊢ (𝜑 → ∀𝑓 ∈ (𝑋𝐻𝑌)(𝑓(⟨𝑋, 𝑋⟩ · 𝑌)( 1 ‘𝑋)) = 𝑓)
33		catlid.f	. 2 ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐻𝑌))
34	3, 32, 33	rspcdva 3566	1 ⊢ (𝜑 → (𝐹(⟨𝑋, 𝑋⟩ · 𝑌)( 1 ‘𝑋)) = 𝐹)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ∀wral 3052  ∃!wreu 3341  {crab 3390  ⟨cop 4574  ‘cfv 6492  ℩crio 7316  (class class class)co 7360  Basecbs 17170  Hom chom 17222  compcco 17223  Catccat 17621  Idccid 17622

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pr 5370

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rmo 3343  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-riota 7317  df-ov 7363  df-cat 17625  df-cid 17626

This theorem is referenced by:  oppccatid  17676  sectcan  17713  monsect  17741  invisoinvl  17748  rcaninv  17752  subccatid  17804  fucidcl  17926  fucrid  17928  invfuc  17935  arwrid  18031  xpccatid  18145  curf2cl  18188  curfuncf  18195  uncfcurf  18196  hofcl  18216  yonedalem3b  18236  bj-endmnd  37648  endmndlem  49502  idepi  49508  upeu2lem  49515  fucorid  49849  precofvalALT  49855  concom  50150