Theorem catrid 17718

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 7435	. . 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 2745	. 2 ⊢ (𝑓 = 𝐹 → ((𝑓(⟨𝑋, 𝑋⟩ · 𝑌)( 1 ‘𝑋)) = 𝑓 ↔ (𝐹(⟨𝑋, 𝑋⟩ · 𝑌)( 1 ‘𝑋)) = 𝐹))
4		oveq2 7436	. . . 4 ⊢ (𝑦 = 𝑌 → (𝑋𝐻𝑦) = (𝑋𝐻𝑌))
5		oveq2 7436	. . . . . 6 ⊢ (𝑦 = 𝑌 → (⟨𝑋, 𝑋⟩ · 𝑦) = (⟨𝑋, 𝑋⟩ · 𝑌))
6	5	oveqd 7445	. . . . 5 ⊢ (𝑦 = 𝑌 → (𝑓(⟨𝑋, 𝑋⟩ · 𝑦)( 1 ‘𝑋)) = (𝑓(⟨𝑋, 𝑋⟩ · 𝑌)( 1 ‘𝑋)))
7	6	eqeq1d 2731	. . . 4 ⊢ (𝑦 = 𝑌 → ((𝑓(⟨𝑋, 𝑋⟩ · 𝑦)( 1 ‘𝑋)) = 𝑓 ↔ (𝑓(⟨𝑋, 𝑋⟩ · 𝑌)( 1 ‘𝑋)) = 𝑓))
8	4, 7	raleqbidv 3339	. . 3 ⊢ (𝑦 = 𝑌 → (∀𝑓 ∈ (𝑋𝐻𝑦)(𝑓(⟨𝑋, 𝑋⟩ · 𝑦)( 1 ‘𝑋)) = 𝑓 ↔ ∀𝑓 ∈ (𝑋𝐻𝑌)(𝑓(⟨𝑋, 𝑋⟩ · 𝑌)( 1 ‘𝑋)) = 𝑓))
9		simpr 483	. . . . . . . 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 3076	. . . . . . 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 4081	. . . . 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 17711	. . . . . 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 17709	. . . . . . 7 ⊢ (𝜑 → ∃!𝑔 ∈ (𝑋𝐻𝑋)∀𝑦 ∈ 𝐵 (∀𝑓 ∈ (𝑦𝐻𝑋)(𝑔(⟨𝑦, 𝑋⟩ · 𝑋)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑋𝐻𝑦)(𝑓(⟨𝑋, 𝑋⟩ · 𝑦)𝑔) = 𝑓))
21		riotacl2 7401	. . . . . . 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 2829	. . . . 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 3987	. . . 4 ⊢ (𝜑 → ( 1 ‘𝑋) ∈ {𝑔 ∈ (𝑋𝐻𝑋) ∣ ∀𝑦 ∈ 𝐵 ∀𝑓 ∈ (𝑋𝐻𝑦)(𝑓(⟨𝑋, 𝑋⟩ · 𝑦)𝑔) = 𝑓})
25		oveq2 7436	. . . . . . . 8 ⊢ (𝑔 = ( 1 ‘𝑋) → (𝑓(⟨𝑋, 𝑋⟩ · 𝑦)𝑔) = (𝑓(⟨𝑋, 𝑋⟩ · 𝑦)( 1 ‘𝑋)))
26	25	eqeq1d 2731	. . . . . . 7 ⊢ (𝑔 = ( 1 ‘𝑋) → ((𝑓(⟨𝑋, 𝑋⟩ · 𝑦)𝑔) = 𝑓 ↔ (𝑓(⟨𝑋, 𝑋⟩ · 𝑦)( 1 ‘𝑋)) = 𝑓))
27	26	2ralbidv 3213	. . . . . 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 3691	. . . . 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 495	. . . 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 3619	. 2 ⊢ (𝜑 → ∀𝑓 ∈ (𝑋𝐻𝑌)(𝑓(⟨𝑋, 𝑋⟩ · 𝑌)( 1 ‘𝑋)) = 𝑓)
33		catlid.f	. 2 ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐻𝑌))
34	3, 32, 33	rspcdva 3619	1 ⊢ (𝜑 → (𝐹(⟨𝑋, 𝑋⟩ · 𝑌)( 1 ‘𝑋)) = 𝐹)

Syntax hints:   → wi 4   ∧ wa 394   = wceq 1534   ∈ wcel 2100  ∀wral 3054  ∃!wreu 3371  {crab 3428  ⟨cop 4640  ‘cfv 6556  ℩crio 7383  (class class class)co 7428  Basecbs 17234  Hom chom 17298  compcco 17299  Catccat 17698  Idccid 17699

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2102  ax-9 2110  ax-10 2133  ax-11 2150  ax-12 2170  ax-ext 2700  ax-rep 5291  ax-sep 5305  ax-nul 5312  ax-pr 5435

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2062  df-mo 2532  df-eu 2561  df-clab 2707  df-cleq 2721  df-clel 2806  df-nfc 2881  df-ne 2934  df-ral 3055  df-rex 3064  df-rmo 3373  df-reu 3374  df-rab 3429  df-v 3474  df-sbc 3787  df-csb 3903  df-dif 3960  df-un 3962  df-in 3964  df-ss 3974  df-nul 4334  df-if 4535  df-sn 4635  df-pr 4637  df-op 4641  df-uni 4917  df-iun 5006  df-br 5155  df-opab 5217  df-mpt 5238  df-id 5582  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-iota 6508  df-fun 6558  df-fn 6559  df-f 6560  df-f1 6561  df-fo 6562  df-f1o 6563  df-fv 6564  df-riota 7384  df-ov 7431  df-cat 17702  df-cid 17703

This theorem is referenced by:  oppccatid  17755  sectcan  17792  monsect  17820  invisoinvl  17827  rcaninv  17831  subccatid  17886  fucidcl  18011  fucrid  18013  invfuc  18020  arwrid  18116  xpccatid  18233  curf2cl  18277  curfuncf  18284  uncfcurf  18285  hofcl  18305  yonedalem3b  18325  bj-endmnd  37062  endmndlem  48407  idepi  48409