Theorem catidcl 17308

Description: Each object in a category has an associated 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)

Assertion

Ref	Expression
catidcl	⊢ (𝜑 → ( 1 ‘𝑋) ∈ (𝑋𝐻𝑋))

Proof of Theorem catidcl

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

Step	Hyp	Ref	Expression
1		catidcl.b	. . 3 ⊢ 𝐵 = (Base‘𝐶)
2		catidcl.h	. . 3 ⊢ 𝐻 = (Hom ‘𝐶)
3		eqid 2738	. . 3 ⊢ (comp‘𝐶) = (comp‘𝐶)
4		catidcl.c	. . 3 ⊢ (𝜑 → 𝐶 ∈ Cat)
5		catidcl.i	. . 3 ⊢ 1 = (Id‘𝐶)
6		catidcl.x	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
7	1, 2, 3, 4, 5, 6	cidval 17303	. 2 ⊢ (𝜑 → ( 1 ‘𝑋) = (℩𝑔 ∈ (𝑋𝐻𝑋)∀𝑦 ∈ 𝐵 (∀𝑓 ∈ (𝑦𝐻𝑋)(𝑔(⟨𝑦, 𝑋⟩(comp‘𝐶)𝑋)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑋𝐻𝑦)(𝑓(⟨𝑋, 𝑋⟩(comp‘𝐶)𝑦)𝑔) = 𝑓)))
8	1, 2, 3, 4, 6	catideu 17301	. . 3 ⊢ (𝜑 → ∃!𝑔 ∈ (𝑋𝐻𝑋)∀𝑦 ∈ 𝐵 (∀𝑓 ∈ (𝑦𝐻𝑋)(𝑔(⟨𝑦, 𝑋⟩(comp‘𝐶)𝑋)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑋𝐻𝑦)(𝑓(⟨𝑋, 𝑋⟩(comp‘𝐶)𝑦)𝑔) = 𝑓))
9		riotacl 7230	. . 3 ⊢ (∃!𝑔 ∈ (𝑋𝐻𝑋)∀𝑦 ∈ 𝐵 (∀𝑓 ∈ (𝑦𝐻𝑋)(𝑔(⟨𝑦, 𝑋⟩(comp‘𝐶)𝑋)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑋𝐻𝑦)(𝑓(⟨𝑋, 𝑋⟩(comp‘𝐶)𝑦)𝑔) = 𝑓) → (℩𝑔 ∈ (𝑋𝐻𝑋)∀𝑦 ∈ 𝐵 (∀𝑓 ∈ (𝑦𝐻𝑋)(𝑔(⟨𝑦, 𝑋⟩(comp‘𝐶)𝑋)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑋𝐻𝑦)(𝑓(⟨𝑋, 𝑋⟩(comp‘𝐶)𝑦)𝑔) = 𝑓)) ∈ (𝑋𝐻𝑋))
10	8, 9	syl 17	. 2 ⊢ (𝜑 → (℩𝑔 ∈ (𝑋𝐻𝑋)∀𝑦 ∈ 𝐵 (∀𝑓 ∈ (𝑦𝐻𝑋)(𝑔(⟨𝑦, 𝑋⟩(comp‘𝐶)𝑋)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑋𝐻𝑦)(𝑓(⟨𝑋, 𝑋⟩(comp‘𝐶)𝑦)𝑔) = 𝑓)) ∈ (𝑋𝐻𝑋))
11	7, 10	eqeltrd 2839	1 ⊢ (𝜑 → ( 1 ‘𝑋) ∈ (𝑋𝐻𝑋))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1539   ∈ wcel 2108  ∀wral 3063  ∃!wreu 3065  ⟨cop 4564  ‘cfv 6418  ℩crio 7211  (class class class)co 7255  Basecbs 16840  Hom chom 16899  compcco 16900  Catccat 17290  Idccid 17291

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rmo 3071  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-riota 7212  df-ov 7258  df-cat 17294  df-cid 17295

This theorem is referenced by:  oppccatid  17347  monsect  17412  sectid  17415  catsubcat  17470  fullsubc  17481  idfucl  17512  cofucl  17519  fthsect  17557  fucidcl  17599  initoid  17632  termoid  17633  idahom  17691  catcisolem  17741  xpccatid  17821  1stfcl  17830  2ndfcl  17831  prfcl  17836  evlfcl  17856  curf1cl  17862  curf2cl  17865  curfcl  17866  curfuncf  17872  uncfcurf  17873  diag12  17878  diag2  17879  curf2ndf  17881  hofcl  17893  yon12  17899  yon2  17900  yonedalem3a  17908  yonedalem3b  17913  yonedainv  17915  bj-endmnd  35416  catprs  46180  endmndlem  46184  idmon  46185  idepi  46186  thincid  46202  functhinclem4  46213