Theorem catlid 17756

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 7424	. . 3 ⊢ (𝑓 = 𝐹 → (( 1 ‘𝑌)(⟨𝑋, 𝑌⟩ · 𝑌)𝑓) = (( 1 ‘𝑌)(⟨𝑋, 𝑌⟩ · 𝑌)𝐹))
2		id 23	. . 3 ⊢ (𝑓 = 𝐹 → 𝑓 = 𝐹)
3	1, 2	eqeq12d 2781	. 2 ⊢ (𝑓 = 𝐹 → ((( 1 ‘𝑌)(⟨𝑋, 𝑌⟩ · 𝑌)𝑓) = 𝑓 ↔ (( 1 ‘𝑌)(⟨𝑋, 𝑌⟩ · 𝑌)𝐹) = 𝐹))
4		oveq1 7423	. . . 4 ⊢ (𝑥 = 𝑋 → (𝑥𝐻𝑌) = (𝑋𝐻𝑌))
5		opeq1 4840	. . . . . . 7 ⊢ (𝑥 = 𝑋 → ⟨𝑥, 𝑌⟩ = ⟨𝑋, 𝑌⟩)
6	5	oveq1d 7431	. . . . . 6 ⊢ (𝑥 = 𝑋 → (⟨𝑥, 𝑌⟩ · 𝑌) = (⟨𝑋, 𝑌⟩ · 𝑌))
7	6	oveqd 7433	. . . . 5 ⊢ (𝑥 = 𝑋 → (( 1 ‘𝑌)(⟨𝑥, 𝑌⟩ · 𝑌)𝑓) = (( 1 ‘𝑌)(⟨𝑋, 𝑌⟩ · 𝑌)𝑓))
8	7	eqeq1d 2767	. . . 4 ⊢ (𝑥 = 𝑋 → ((( 1 ‘𝑌)(⟨𝑥, 𝑌⟩ · 𝑌)𝑓) = 𝑓 ↔ (( 1 ‘𝑌)(⟨𝑋, 𝑌⟩ · 𝑌)𝑓) = 𝑓))
9	4, 8	raleqbidv 3340	. . 3 ⊢ (𝑥 = 𝑋 → (∀𝑓 ∈ (𝑥𝐻𝑌)(( 1 ‘𝑌)(⟨𝑥, 𝑌⟩ · 𝑌)𝑓) = 𝑓 ↔ ∀𝑓 ∈ (𝑋𝐻𝑌)(( 1 ‘𝑌)(⟨𝑋, 𝑌⟩ · 𝑌)𝑓) = 𝑓))
10		simpl 488	. . . . . . . 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 3104	. . . . . . 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 4031	. . . . 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 17750	. . . . . 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 17748	. . . . . . 7 ⊢ (𝜑 → ∃!𝑔 ∈ (𝑌𝐻𝑌)∀𝑥 ∈ 𝐵 (∀𝑓 ∈ (𝑥𝐻𝑌)(𝑔(⟨𝑥, 𝑌⟩ · 𝑌)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑌𝐻𝑥)(𝑓(⟨𝑌, 𝑌⟩ · 𝑥)𝑔) = 𝑓))
22		riotacl2 7389	. . . . . . 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 18	. . . . . 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 2865	. . . . 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 3936	. . . 4 ⊢ (𝜑 → ( 1 ‘𝑌) ∈ {𝑔 ∈ (𝑌𝐻𝑌) ∣ ∀𝑥 ∈ 𝐵 ∀𝑓 ∈ (𝑥𝐻𝑌)(𝑔(⟨𝑥, 𝑌⟩ · 𝑌)𝑓) = 𝑓})
26		oveq1 7423	. . . . . . . 8 ⊢ (𝑔 = ( 1 ‘𝑌) → (𝑔(⟨𝑥, 𝑌⟩ · 𝑌)𝑓) = (( 1 ‘𝑌)(⟨𝑥, 𝑌⟩ · 𝑌)𝑓))
27	26	eqeq1d 2767	. . . . . . 7 ⊢ (𝑔 = ( 1 ‘𝑌) → ((𝑔(⟨𝑥, 𝑌⟩ · 𝑌)𝑓) = 𝑓 ↔ (( 1 ‘𝑌)(⟨𝑥, 𝑌⟩ · 𝑌)𝑓) = 𝑓))
28	27	2ralbidv 3231	. . . . . 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 3652	. . . . 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 503	. . . 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 18	. . 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 3584	. 2 ⊢ (𝜑 → ∀𝑓 ∈ (𝑋𝐻𝑌)(( 1 ‘𝑌)(⟨𝑋, 𝑌⟩ · 𝑌)𝑓) = 𝑓)
34		catlid.f	. 2 ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐻𝑌))
35	3, 33, 34	rspcdva 3584	1 ⊢ (𝜑 → (( 1 ‘𝑌)(⟨𝑋, 𝑌⟩ · 𝑌)𝐹) = 𝐹)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2146  ∀wral 3081  ∃!wreu 3369  {crab 3418  ⟨cop 4597  ‘cfv 6540  ℩crio 7372  (class class class)co 7416  Basecbs 17286  Hom chom 17338  compcco 17339  Catccat 17737  Idccid 17738

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-rep 5240  ax-sep 5259  ax-nul 5271  ax-pr 5406

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-rmo 3371  df-reu 3372  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-iun 4960  df-br 5112  df-opab 5176  df-mpt 5195  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7373  df-ov 7419  df-cat 17741  df-cid 17742

This theorem is used by:  oppccatid  17792  sectcan  17829  sectco  17830  sectmon  17856  monsect  17857  sectid  17860  invisoinvl  17864  subccatid  17920  fucidcl  18042  fuclid  18043  invfuc  18051  arwlid  18146  xpccatid  18261  evlfcl  18295  curf1cl  18301  curf2cl  18304  curfcl  18305  curfuncf  18311  uncfcurf  18312  hofcl  18332  yon12  18338  yon2  18339  yonedalem3b  18352  yonedainv  18354  bj-endmnd  37995  endmndlem  49826  idmon  49831  discsubc  49875  upciclem3  49979  fucoid  50159  fucolid  50172  coccom  50475