Theorem catlid 17706

Description: Left 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
catlid	⊢ (𝜑 → (( 1 ‘𝑌)(⟨𝑋, 𝑌⟩ · 𝑌)𝐹) = 𝐹)

Proof of Theorem catlid

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

Step	Hyp	Ref	Expression
1		oveq2 7399	. . 3 ⊢ (𝑓 = 𝐹 → (( 1 ‘𝑌)(⟨𝑋, 𝑌⟩ · 𝑌)𝑓) = (( 1 ‘𝑌)(⟨𝑋, 𝑌⟩ · 𝑌)𝐹))
2		id 22	. . 3 ⊢ (𝑓 = 𝐹 → 𝑓 = 𝐹)
3	1, 2	eqeq12d 2777	. 2 ⊢ (𝑓 = 𝐹 → ((( 1 ‘𝑌)(⟨𝑋, 𝑌⟩ · 𝑌)𝑓) = 𝑓 ↔ (( 1 ‘𝑌)(⟨𝑋, 𝑌⟩ · 𝑌)𝐹) = 𝐹))
4		oveq1 7398	. . . 4 ⊢ (𝑥 = 𝑋 → (𝑥𝐻𝑌) = (𝑋𝐻𝑌))
5		opeq1 4828	. . . . . . 7 ⊢ (𝑥 = 𝑋 → ⟨𝑥, 𝑌⟩ = ⟨𝑋, 𝑌⟩)
6	5	oveq1d 7406	. . . . . 6 ⊢ (𝑥 = 𝑋 → (⟨𝑥, 𝑌⟩ · 𝑌) = (⟨𝑋, 𝑌⟩ · 𝑌))
7	6	oveqd 7408	. . . . 5 ⊢ (𝑥 = 𝑋 → (( 1 ‘𝑌)(⟨𝑥, 𝑌⟩ · 𝑌)𝑓) = (( 1 ‘𝑌)(⟨𝑋, 𝑌⟩ · 𝑌)𝑓))
8	7	eqeq1d 2763	. . . 4 ⊢ (𝑥 = 𝑋 → ((( 1 ‘𝑌)(⟨𝑥, 𝑌⟩ · 𝑌)𝑓) = 𝑓 ↔ (( 1 ‘𝑌)(⟨𝑋, 𝑌⟩ · 𝑌)𝑓) = 𝑓))
9	4, 8	raleqbidv 3335	. . 3 ⊢ (𝑥 = 𝑋 → (∀𝑓 ∈ (𝑥𝐻𝑌)(( 1 ‘𝑌)(⟨𝑥, 𝑌⟩ · 𝑌)𝑓) = 𝑓 ↔ ∀𝑓 ∈ (𝑋𝐻𝑌)(( 1 ‘𝑌)(⟨𝑋, 𝑌⟩ · 𝑌)𝑓) = 𝑓))
10		simpl 486	. . . . . . . 8 ⊢ ((∀𝑓 ∈ (𝑥𝐻𝑌)(𝑔(⟨𝑥, 𝑌⟩ · 𝑌)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑌𝐻𝑥)(𝑓(⟨𝑌, 𝑌⟩ · 𝑥)𝑔) = 𝑓) → ∀𝑓 ∈ (𝑥𝐻𝑌)(𝑔(⟨𝑥, 𝑌⟩ · 𝑌)𝑓) = 𝑓)
11	10	ralimi 3098	. . . . . . 7 ⊢ (∀𝑥 ∈ 𝐵 (∀𝑓 ∈ (𝑥𝐻𝑌)(𝑔(⟨𝑥, 𝑌⟩ · 𝑌)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑌𝐻𝑥)(𝑓(⟨𝑌, 𝑌⟩ · 𝑥)𝑔) = 𝑓) → ∀𝑥 ∈ 𝐵 ∀𝑓 ∈ (𝑥𝐻𝑌)(𝑔(⟨𝑥, 𝑌⟩ · 𝑌)𝑓) = 𝑓)
12	11	a1i 11	. . . . . 6 ⊢ (𝑔 ∈ (𝑌𝐻𝑌) → (∀𝑥 ∈ 𝐵 (∀𝑓 ∈ (𝑥𝐻𝑌)(𝑔(⟨𝑥, 𝑌⟩ · 𝑌)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑌𝐻𝑥)(𝑓(⟨𝑌, 𝑌⟩ · 𝑥)𝑔) = 𝑓) → ∀𝑥 ∈ 𝐵 ∀𝑓 ∈ (𝑥𝐻𝑌)(𝑔(⟨𝑥, 𝑌⟩ · 𝑌)𝑓) = 𝑓))
13	12	ss2rabi 4027	. . . . 5 ⊢ {𝑔 ∈ (𝑌𝐻𝑌) ∣ ∀𝑥 ∈ 𝐵 (∀𝑓 ∈ (𝑥𝐻𝑌)(𝑔(⟨𝑥, 𝑌⟩ · 𝑌)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑌𝐻𝑥)(𝑓(⟨𝑌, 𝑌⟩ · 𝑥)𝑔) = 𝑓)} ⊆ {𝑔 ∈ (𝑌𝐻𝑌) ∣ ∀𝑥 ∈ 𝐵 ∀𝑓 ∈ (𝑥𝐻𝑌)(𝑔(⟨𝑥, 𝑌⟩ · 𝑌)𝑓) = 𝑓}
14		catidcl.b	. . . . . . 7 ⊢ 𝐵 = (Base‘𝐶)
15		catidcl.h	. . . . . . 7 ⊢ 𝐻 = (Hom ‘𝐶)
16		catlid.o	. . . . . . 7 ⊢ · = (comp‘𝐶)
17		catidcl.c	. . . . . . 7 ⊢ (𝜑 → 𝐶 ∈ Cat)
18		catidcl.i	. . . . . . 7 ⊢ 1 = (Id‘𝐶)
19		catlid.y	. . . . . . 7 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
20	14, 15, 16, 17, 18, 19	cidval 17700	. . . . . 6 ⊢ (𝜑 → ( 1 ‘𝑌) = (℩𝑔 ∈ (𝑌𝐻𝑌)∀𝑥 ∈ 𝐵 (∀𝑓 ∈ (𝑥𝐻𝑌)(𝑔(⟨𝑥, 𝑌⟩ · 𝑌)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑌𝐻𝑥)(𝑓(⟨𝑌, 𝑌⟩ · 𝑥)𝑔) = 𝑓)))
21	14, 15, 16, 17, 19	catideu 17698	. . . . . . 7 ⊢ (𝜑 → ∃!𝑔 ∈ (𝑌𝐻𝑌)∀𝑥 ∈ 𝐵 (∀𝑓 ∈ (𝑥𝐻𝑌)(𝑔(⟨𝑥, 𝑌⟩ · 𝑌)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑌𝐻𝑥)(𝑓(⟨𝑌, 𝑌⟩ · 𝑥)𝑔) = 𝑓))
22		riotacl2 7364	. . . . . . 7 ⊢ (∃!𝑔 ∈ (𝑌𝐻𝑌)∀𝑥 ∈ 𝐵 (∀𝑓 ∈ (𝑥𝐻𝑌)(𝑔(⟨𝑥, 𝑌⟩ · 𝑌)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑌𝐻𝑥)(𝑓(⟨𝑌, 𝑌⟩ · 𝑥)𝑔) = 𝑓) → (℩𝑔 ∈ (𝑌𝐻𝑌)∀𝑥 ∈ 𝐵 (∀𝑓 ∈ (𝑥𝐻𝑌)(𝑔(⟨𝑥, 𝑌⟩ · 𝑌)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑌𝐻𝑥)(𝑓(⟨𝑌, 𝑌⟩ · 𝑥)𝑔) = 𝑓)) ∈ {𝑔 ∈ (𝑌𝐻𝑌) ∣ ∀𝑥 ∈ 𝐵 (∀𝑓 ∈ (𝑥𝐻𝑌)(𝑔(⟨𝑥, 𝑌⟩ · 𝑌)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑌𝐻𝑥)(𝑓(⟨𝑌, 𝑌⟩ · 𝑥)𝑔) = 𝑓)})
23	21, 22	syl 17	. . . . . 6 ⊢ (𝜑 → (℩𝑔 ∈ (𝑌𝐻𝑌)∀𝑥 ∈ 𝐵 (∀𝑓 ∈ (𝑥𝐻𝑌)(𝑔(⟨𝑥, 𝑌⟩ · 𝑌)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑌𝐻𝑥)(𝑓(⟨𝑌, 𝑌⟩ · 𝑥)𝑔) = 𝑓)) ∈ {𝑔 ∈ (𝑌𝐻𝑌) ∣ ∀𝑥 ∈ 𝐵 (∀𝑓 ∈ (𝑥𝐻𝑌)(𝑔(⟨𝑥, 𝑌⟩ · 𝑌)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑌𝐻𝑥)(𝑓(⟨𝑌, 𝑌⟩ · 𝑥)𝑔) = 𝑓)})
24	20, 23	eqeltrd 2861	. . . . 5 ⊢ (𝜑 → ( 1 ‘𝑌) ∈ {𝑔 ∈ (𝑌𝐻𝑌) ∣ ∀𝑥 ∈ 𝐵 (∀𝑓 ∈ (𝑥𝐻𝑌)(𝑔(⟨𝑥, 𝑌⟩ · 𝑌)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑌𝐻𝑥)(𝑓(⟨𝑌, 𝑌⟩ · 𝑥)𝑔) = 𝑓)})
25	13, 24	sselid 3932	. . . 4 ⊢ (𝜑 → ( 1 ‘𝑌) ∈ {𝑔 ∈ (𝑌𝐻𝑌) ∣ ∀𝑥 ∈ 𝐵 ∀𝑓 ∈ (𝑥𝐻𝑌)(𝑔(⟨𝑥, 𝑌⟩ · 𝑌)𝑓) = 𝑓})
26		oveq1 7398	. . . . . . . 8 ⊢ (𝑔 = ( 1 ‘𝑌) → (𝑔(⟨𝑥, 𝑌⟩ · 𝑌)𝑓) = (( 1 ‘𝑌)(⟨𝑥, 𝑌⟩ · 𝑌)𝑓))
27	26	eqeq1d 2763	. . . . . . 7 ⊢ (𝑔 = ( 1 ‘𝑌) → ((𝑔(⟨𝑥, 𝑌⟩ · 𝑌)𝑓) = 𝑓 ↔ (( 1 ‘𝑌)(⟨𝑥, 𝑌⟩ · 𝑌)𝑓) = 𝑓))
28	27	2ralbidv 3225	. . . . . 6 ⊢ (𝑔 = ( 1 ‘𝑌) → (∀𝑥 ∈ 𝐵 ∀𝑓 ∈ (𝑥𝐻𝑌)(𝑔(⟨𝑥, 𝑌⟩ · 𝑌)𝑓) = 𝑓 ↔ ∀𝑥 ∈ 𝐵 ∀𝑓 ∈ (𝑥𝐻𝑌)(( 1 ‘𝑌)(⟨𝑥, 𝑌⟩ · 𝑌)𝑓) = 𝑓))
29	28	elrab 3649	. . . . 5 ⊢ (( 1 ‘𝑌) ∈ {𝑔 ∈ (𝑌𝐻𝑌) ∣ ∀𝑥 ∈ 𝐵 ∀𝑓 ∈ (𝑥𝐻𝑌)(𝑔(⟨𝑥, 𝑌⟩ · 𝑌)𝑓) = 𝑓} ↔ (( 1 ‘𝑌) ∈ (𝑌𝐻𝑌) ∧ ∀𝑥 ∈ 𝐵 ∀𝑓 ∈ (𝑥𝐻𝑌)(( 1 ‘𝑌)(⟨𝑥, 𝑌⟩ · 𝑌)𝑓) = 𝑓))
30	29	simprbi 501	. . . 4 ⊢ (( 1 ‘𝑌) ∈ {𝑔 ∈ (𝑌𝐻𝑌) ∣ ∀𝑥 ∈ 𝐵 ∀𝑓 ∈ (𝑥𝐻𝑌)(𝑔(⟨𝑥, 𝑌⟩ · 𝑌)𝑓) = 𝑓} → ∀𝑥 ∈ 𝐵 ∀𝑓 ∈ (𝑥𝐻𝑌)(( 1 ‘𝑌)(⟨𝑥, 𝑌⟩ · 𝑌)𝑓) = 𝑓)
31	25, 30	syl 17	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑓 ∈ (𝑥𝐻𝑌)(( 1 ‘𝑌)(⟨𝑥, 𝑌⟩ · 𝑌)𝑓) = 𝑓)
32		catidcl.x	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
33	9, 31, 32	rspcdva 3581	. 2 ⊢ (𝜑 → ∀𝑓 ∈ (𝑋𝐻𝑌)(( 1 ‘𝑌)(⟨𝑋, 𝑌⟩ · 𝑌)𝑓) = 𝑓)
34		catlid.f	. 2 ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐻𝑌))
35	3, 33, 34	rspcdva 3581	1 ⊢ (𝜑 → (( 1 ‘𝑌)(⟨𝑋, 𝑌⟩ · 𝑌)𝐹) = 𝐹)

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1559   ∈ wcel 2141  ∀wral 3075  ∃!wreu 3364  {crab 3413  ⟨cop 4585  ‘cfv 6516  ℩crio 7347  (class class class)co 7391  Basecbs 17236  Hom chom 17288  compcco 17289  Catccat 17687  Idccid 17688

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5224  ax-sep 5243  ax-nul 5253  ax-pr 5387

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4478  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-iun 4948  df-br 5098  df-opab 5160  df-mpt 5179  df-id 5538  df-xp 5649  df-rel 5650  df-cnv 5651  df-co 5652  df-dm 5653  df-rn 5654  df-res 5655  df-ima 5656  df-iota 6472  df-fun 6518  df-fn 6519  df-f 6520  df-f1 6521  df-fo 6522  df-f1o 6523  df-fv 6524  df-riota 7348  df-ov 7394  df-cat 17691  df-cid 17692

This theorem is referenced by:  oppccatid  17742  sectcan  17779  sectco  17780  sectmon  17806  monsect  17807  sectid  17810  invisoinvl  17814  subccatid  17870  fucidcl  17992  fuclid  17993  invfuc  18001  arwlid  18096  xpccatid  18211  evlfcl  18245  curf1cl  18251  curf2cl  18254  curfcl  18255  curfuncf  18261  uncfcurf  18262  hofcl  18282  yon12  18288  yon2  18289  yonedalem3b  18302  yonedainv  18304  bj-endmnd  37771  endmndlem  49597  idmon  49602  discsubc  49646  upciclem3  49750  fucoid  49930  fucolid  49943  coccom  50246