Theorem catidcl 17639

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 2739	. . 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 17634	. 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 17632	. . 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 7330	. . 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 396   = wceq 1547   ∈ wcel 2119  ∀wral 3053  ∃!wreu 3342  ⟨cop 4561  ‘cfv 6485  ℩crio 7312  (class class class)co 7356  Basecbs 17170  Hom chom 17222  compcco 17223  Catccat 17621  Idccid 17622

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-rep 5199  ax-sep 5218  ax-nul 5228  ax-pr 5362

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-ral 3054  df-rex 3064  df-rmo 3344  df-reu 3345  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-iun 4923  df-br 5073  df-opab 5135  df-mpt 5154  df-id 5513  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-iota 6441  df-fun 6487  df-fn 6488  df-f 6489  df-f1 6490  df-fo 6491  df-f1o 6492  df-fv 6493  df-riota 7313  df-ov 7359  df-cat 17625  df-cid 17626

This theorem is referenced by:  oppccatid  17676  monsect  17741  sectid  17744  catsubcat  17797  fullsubc  17808  idfucl  17839  cofucl  17846  fthsect  17885  fucidcl  17926  initoid  17959  termoid  17960  idahom  18018  catcisolem  18068  xpccatid  18145  1stfcl  18154  2ndfcl  18155  prfcl  18160  evlfcl  18179  curf1cl  18185  curf2cl  18188  curfcl  18189  curfuncf  18195  uncfcurf  18196  diag12  18201  diag2  18202  curf2ndf  18204  hofcl  18216  yon12  18222  yon2  18223  yonedalem3a  18231  yonedalem3b  18236  yonedainv  18238  bj-endmnd  37678  catprs  49501  endmndlem  49505  idmon  49510  idepi  49511  discsubc  49554  imaid  49644  upciclem3  49658  swapfid  49769  tposcurf12  49788  tposcurf2  49790  fucoid  49838  thincid  49922  functhinclem4  49937  termchomn0  49974  idfudiag1  50015  termcarweu  50018  arweutermc  50020