Theorem catlid 17618

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 7376	. . 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 2753	. 2 ⊢ (𝑓 = 𝐹 → ((( 1 ‘𝑌)(⟨𝑋, 𝑌⟩ · 𝑌)𝑓) = 𝑓 ↔ (( 1 ‘𝑌)(⟨𝑋, 𝑌⟩ · 𝑌)𝐹) = 𝐹))
4		oveq1 7375	. . . 4 ⊢ (𝑥 = 𝑋 → (𝑥𝐻𝑌) = (𝑋𝐻𝑌))
5		opeq1 4831	. . . . . . 7 ⊢ (𝑥 = 𝑋 → ⟨𝑥, 𝑌⟩ = ⟨𝑋, 𝑌⟩)
6	5	oveq1d 7383	. . . . . 6 ⊢ (𝑥 = 𝑋 → (⟨𝑥, 𝑌⟩ · 𝑌) = (⟨𝑋, 𝑌⟩ · 𝑌))
7	6	oveqd 7385	. . . . 5 ⊢ (𝑥 = 𝑋 → (( 1 ‘𝑌)(⟨𝑥, 𝑌⟩ · 𝑌)𝑓) = (( 1 ‘𝑌)(⟨𝑋, 𝑌⟩ · 𝑌)𝑓))
8	7	eqeq1d 2739	. . . 4 ⊢ (𝑥 = 𝑋 → ((( 1 ‘𝑌)(⟨𝑥, 𝑌⟩ · 𝑌)𝑓) = 𝑓 ↔ (( 1 ‘𝑌)(⟨𝑋, 𝑌⟩ · 𝑌)𝑓) = 𝑓))
9	4, 8	raleqbidv 3318	. . 3 ⊢ (𝑥 = 𝑋 → (∀𝑓 ∈ (𝑥𝐻𝑌)(( 1 ‘𝑌)(⟨𝑥, 𝑌⟩ · 𝑌)𝑓) = 𝑓 ↔ ∀𝑓 ∈ (𝑋𝐻𝑌)(( 1 ‘𝑌)(⟨𝑋, 𝑌⟩ · 𝑌)𝑓) = 𝑓))
10		simpl 482	. . . . . . . 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 3075	. . . . . . 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 4030	. . . . 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 17612	. . . . . 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 17610	. . . . . . 7 ⊢ (𝜑 → ∃!𝑔 ∈ (𝑌𝐻𝑌)∀𝑥 ∈ 𝐵 (∀𝑓 ∈ (𝑥𝐻𝑌)(𝑔(⟨𝑥, 𝑌⟩ · 𝑌)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑌𝐻𝑥)(𝑓(⟨𝑌, 𝑌⟩ · 𝑥)𝑔) = 𝑓))
22		riotacl2 7341	. . . . . . 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 2837	. . . . 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 3933	. . . 4 ⊢ (𝜑 → ( 1 ‘𝑌) ∈ {𝑔 ∈ (𝑌𝐻𝑌) ∣ ∀𝑥 ∈ 𝐵 ∀𝑓 ∈ (𝑥𝐻𝑌)(𝑔(⟨𝑥, 𝑌⟩ · 𝑌)𝑓) = 𝑓})
26		oveq1 7375	. . . . . . . 8 ⊢ (𝑔 = ( 1 ‘𝑌) → (𝑔(⟨𝑥, 𝑌⟩ · 𝑌)𝑓) = (( 1 ‘𝑌)(⟨𝑥, 𝑌⟩ · 𝑌)𝑓))
27	26	eqeq1d 2739	. . . . . . 7 ⊢ (𝑔 = ( 1 ‘𝑌) → ((𝑔(⟨𝑥, 𝑌⟩ · 𝑌)𝑓) = 𝑓 ↔ (( 1 ‘𝑌)(⟨𝑥, 𝑌⟩ · 𝑌)𝑓) = 𝑓))
28	27	2ralbidv 3202	. . . . . 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 3648	. . . . 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 497	. . . 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 3579	. 2 ⊢ (𝜑 → ∀𝑓 ∈ (𝑋𝐻𝑌)(( 1 ‘𝑌)(⟨𝑋, 𝑌⟩ · 𝑌)𝑓) = 𝑓)
34		catlid.f	. 2 ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐻𝑌))
35	3, 33, 34	rspcdva 3579	1 ⊢ (𝜑 → (( 1 ‘𝑌)(⟨𝑋, 𝑌⟩ · 𝑌)𝐹) = 𝐹)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ∀wral 3052  ∃!wreu 3350  {crab 3401  ⟨cop 4588  ‘cfv 6500  ℩crio 7324  (class class class)co 7368  Basecbs 17148  Hom chom 17200  compcco 17201  Catccat 17599  Idccid 17600

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 5226  ax-sep 5243  ax-nul 5253  ax-pr 5379

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 3352  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-riota 7325  df-ov 7371  df-cat 17603  df-cid 17604

This theorem is referenced by:  oppccatid  17654  sectcan  17691  sectco  17692  sectmon  17718  monsect  17719  sectid  17722  invisoinvl  17726  subccatid  17782  fucidcl  17904  fuclid  17905  invfuc  17913  arwlid  18008  xpccatid  18123  evlfcl  18157  curf1cl  18163  curf2cl  18166  curfcl  18167  curfuncf  18173  uncfcurf  18174  hofcl  18194  yon12  18200  yon2  18201  yonedalem3b  18214  yonedainv  18216  bj-endmnd  37570  endmndlem  49371  idmon  49376  discsubc  49420  upciclem3  49524  fucoid  49704  fucolid  49717  coccom  50020