Theorem catlid 17728

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 7439	. . 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 2751	. 2 ⊢ (𝑓 = 𝐹 → ((( 1 ‘𝑌)(⟨𝑋, 𝑌⟩ · 𝑌)𝑓) = 𝑓 ↔ (( 1 ‘𝑌)(⟨𝑋, 𝑌⟩ · 𝑌)𝐹) = 𝐹))
4		oveq1 7438	. . . 4 ⊢ (𝑥 = 𝑋 → (𝑥𝐻𝑌) = (𝑋𝐻𝑌))
5		opeq1 4878	. . . . . . 7 ⊢ (𝑥 = 𝑋 → ⟨𝑥, 𝑌⟩ = ⟨𝑋, 𝑌⟩)
6	5	oveq1d 7446	. . . . . 6 ⊢ (𝑥 = 𝑋 → (⟨𝑥, 𝑌⟩ · 𝑌) = (⟨𝑋, 𝑌⟩ · 𝑌))
7	6	oveqd 7448	. . . . 5 ⊢ (𝑥 = 𝑋 → (( 1 ‘𝑌)(⟨𝑥, 𝑌⟩ · 𝑌)𝑓) = (( 1 ‘𝑌)(⟨𝑋, 𝑌⟩ · 𝑌)𝑓))
8	7	eqeq1d 2737	. . . 4 ⊢ (𝑥 = 𝑋 → ((( 1 ‘𝑌)(⟨𝑥, 𝑌⟩ · 𝑌)𝑓) = 𝑓 ↔ (( 1 ‘𝑌)(⟨𝑋, 𝑌⟩ · 𝑌)𝑓) = 𝑓))
9	4, 8	raleqbidv 3344	. . 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 3081	. . . . . . 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 4087	. . . . 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 17722	. . . . . 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 17720	. . . . . . 7 ⊢ (𝜑 → ∃!𝑔 ∈ (𝑌𝐻𝑌)∀𝑥 ∈ 𝐵 (∀𝑓 ∈ (𝑥𝐻𝑌)(𝑔(⟨𝑥, 𝑌⟩ · 𝑌)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑌𝐻𝑥)(𝑓(⟨𝑌, 𝑌⟩ · 𝑥)𝑔) = 𝑓))
22		riotacl2 7404	. . . . . . 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 2839	. . . . 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 3993	. . . 4 ⊢ (𝜑 → ( 1 ‘𝑌) ∈ {𝑔 ∈ (𝑌𝐻𝑌) ∣ ∀𝑥 ∈ 𝐵 ∀𝑓 ∈ (𝑥𝐻𝑌)(𝑔(⟨𝑥, 𝑌⟩ · 𝑌)𝑓) = 𝑓})
26		oveq1 7438	. . . . . . . 8 ⊢ (𝑔 = ( 1 ‘𝑌) → (𝑔(⟨𝑥, 𝑌⟩ · 𝑌)𝑓) = (( 1 ‘𝑌)(⟨𝑥, 𝑌⟩ · 𝑌)𝑓))
27	26	eqeq1d 2737	. . . . . . 7 ⊢ (𝑔 = ( 1 ‘𝑌) → ((𝑔(⟨𝑥, 𝑌⟩ · 𝑌)𝑓) = 𝑓 ↔ (( 1 ‘𝑌)(⟨𝑥, 𝑌⟩ · 𝑌)𝑓) = 𝑓))
28	27	2ralbidv 3219	. . . . . 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 3695	. . . . 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 3623	. 2 ⊢ (𝜑 → ∀𝑓 ∈ (𝑋𝐻𝑌)(( 1 ‘𝑌)(⟨𝑋, 𝑌⟩ · 𝑌)𝑓) = 𝑓)
34		catlid.f	. 2 ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐻𝑌))
35	3, 33, 34	rspcdva 3623	1 ⊢ (𝜑 → (( 1 ‘𝑌)(⟨𝑋, 𝑌⟩ · 𝑌)𝐹) = 𝐹)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1537   ∈ wcel 2106  ∀wral 3059  ∃!wreu 3376  {crab 3433  ⟨cop 4637  ‘cfv 6563  ℩crio 7387  (class class class)co 7431  Basecbs 17245  Hom chom 17309  compcco 17310  Catccat 17709  Idccid 17710

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pr 5438

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-cat 17713  df-cid 17714

This theorem is referenced by:  oppccatid  17766  sectcan  17803  sectco  17804  sectmon  17830  monsect  17831  sectid  17834  invisoinvl  17838  subccatid  17897  fucidcl  18022  fuclid  18023  invfuc  18031  arwlid  18126  xpccatid  18244  evlfcl  18279  curf1cl  18285  curf2cl  18288  curfcl  18289  curfuncf  18295  uncfcurf  18296  hofcl  18316  yon12  18322  yon2  18323  yonedalem3b  18336  yonedainv  18338  bj-endmnd  37301  endmndlem  48804  idmon  48805  upciclem3  48814