Theorem catlid 17597

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 7363	. . 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 2749	. 2 ⊢ (𝑓 = 𝐹 → ((( 1 ‘𝑌)(⟨𝑋, 𝑌⟩ · 𝑌)𝑓) = 𝑓 ↔ (( 1 ‘𝑌)(⟨𝑋, 𝑌⟩ · 𝑌)𝐹) = 𝐹))
4		oveq1 7362	. . . 4 ⊢ (𝑥 = 𝑋 → (𝑥𝐻𝑌) = (𝑋𝐻𝑌))
5		opeq1 4826	. . . . . . 7 ⊢ (𝑥 = 𝑋 → ⟨𝑥, 𝑌⟩ = ⟨𝑋, 𝑌⟩)
6	5	oveq1d 7370	. . . . . 6 ⊢ (𝑥 = 𝑋 → (⟨𝑥, 𝑌⟩ · 𝑌) = (⟨𝑋, 𝑌⟩ · 𝑌))
7	6	oveqd 7372	. . . . 5 ⊢ (𝑥 = 𝑋 → (( 1 ‘𝑌)(⟨𝑥, 𝑌⟩ · 𝑌)𝑓) = (( 1 ‘𝑌)(⟨𝑋, 𝑌⟩ · 𝑌)𝑓))
8	7	eqeq1d 2735	. . . 4 ⊢ (𝑥 = 𝑋 → ((( 1 ‘𝑌)(⟨𝑥, 𝑌⟩ · 𝑌)𝑓) = 𝑓 ↔ (( 1 ‘𝑌)(⟨𝑋, 𝑌⟩ · 𝑌)𝑓) = 𝑓))
9	4, 8	raleqbidv 3313	. . 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 3070	. . . . . . 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 4025	. . . . 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 17591	. . . . . 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 17589	. . . . . . 7 ⊢ (𝜑 → ∃!𝑔 ∈ (𝑌𝐻𝑌)∀𝑥 ∈ 𝐵 (∀𝑓 ∈ (𝑥𝐻𝑌)(𝑔(⟨𝑥, 𝑌⟩ · 𝑌)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑌𝐻𝑥)(𝑓(⟨𝑌, 𝑌⟩ · 𝑥)𝑔) = 𝑓))
22		riotacl2 7328	. . . . . . 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 2833	. . . . 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 3928	. . . 4 ⊢ (𝜑 → ( 1 ‘𝑌) ∈ {𝑔 ∈ (𝑌𝐻𝑌) ∣ ∀𝑥 ∈ 𝐵 ∀𝑓 ∈ (𝑥𝐻𝑌)(𝑔(⟨𝑥, 𝑌⟩ · 𝑌)𝑓) = 𝑓})
26		oveq1 7362	. . . . . . . 8 ⊢ (𝑔 = ( 1 ‘𝑌) → (𝑔(⟨𝑥, 𝑌⟩ · 𝑌)𝑓) = (( 1 ‘𝑌)(⟨𝑥, 𝑌⟩ · 𝑌)𝑓))
27	26	eqeq1d 2735	. . . . . . 7 ⊢ (𝑔 = ( 1 ‘𝑌) → ((𝑔(⟨𝑥, 𝑌⟩ · 𝑌)𝑓) = 𝑓 ↔ (( 1 ‘𝑌)(⟨𝑥, 𝑌⟩ · 𝑌)𝑓) = 𝑓))
28	27	2ralbidv 3197	. . . . . 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 3643	. . . . 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 496	. . . 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 3574	. 2 ⊢ (𝜑 → ∀𝑓 ∈ (𝑋𝐻𝑌)(( 1 ‘𝑌)(⟨𝑋, 𝑌⟩ · 𝑌)𝑓) = 𝑓)
34		catlid.f	. 2 ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐻𝑌))
35	3, 33, 34	rspcdva 3574	1 ⊢ (𝜑 → (( 1 ‘𝑌)(⟨𝑋, 𝑌⟩ · 𝑌)𝐹) = 𝐹)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2113  ∀wral 3048  ∃!wreu 3345  {crab 3396  ⟨cop 4583  ‘cfv 6489  ℩crio 7311  (class class class)co 7355  Basecbs 17127  Hom chom 17179  compcco 17180  Catccat 17578  Idccid 17579

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-rep 5221  ax-sep 5238  ax-nul 5248  ax-pr 5374

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-ral 3049  df-rex 3058  df-rmo 3347  df-reu 3348  df-rab 3397  df-v 3439  df-sbc 3738  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4283  df-if 4477  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4861  df-iun 4945  df-br 5096  df-opab 5158  df-mpt 5177  df-id 5516  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-iota 6445  df-fun 6491  df-fn 6492  df-f 6493  df-f1 6494  df-fo 6495  df-f1o 6496  df-fv 6497  df-riota 7312  df-ov 7358  df-cat 17582  df-cid 17583

This theorem is referenced by:  oppccatid  17633  sectcan  17670  sectco  17671  sectmon  17697  monsect  17698  sectid  17701  invisoinvl  17705  subccatid  17761  fucidcl  17883  fuclid  17884  invfuc  17892  arwlid  17987  xpccatid  18102  evlfcl  18136  curf1cl  18142  curf2cl  18145  curfcl  18146  curfuncf  18152  uncfcurf  18153  hofcl  18173  yon12  18179  yon2  18180  yonedalem3b  18193  yonedainv  18195  bj-endmnd  37435  endmndlem  49176  idmon  49181  discsubc  49225  upciclem3  49329  fucoid  49509  fucolid  49522  coccom  49825