Theorem catidcl 17733

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 2763	. . 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 17728	. 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 17726	. . 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 7384	. . 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 18	. 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 2863	1 ⊢ (𝜑 → ( 1 ‘𝑋) ∈ (𝑋𝐻𝑋))

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1570   ∈ wcel 2143  ∀wral 3079  ∃!wreu 3367  ⟨cop 4595  ‘cfv 6536  ℩crio 7366  (class class class)co 7410  Basecbs 17264  Hom chom 17316  compcco 17317  Catccat 17715  Idccid 17716

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5238  ax-sep 5257  ax-nul 5269  ax-pr 5404

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-iun 4958  df-br 5110  df-opab 5174  df-mpt 5193  df-id 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7367  df-ov 7413  df-cat 17719  df-cid 17720

This theorem is referenced by:  oppccatid  17770  monsect  17835  sectid  17838  catsubcat  17891  fullsubc  17902  idfucl  17933  cofucl  17940  fthsect  17979  fucidcl  18020  initoid  18053  termoid  18054  idahom  18112  catcisolem  18162  xpccatid  18239  1stfcl  18248  2ndfcl  18249  prfcl  18254  evlfcl  18273  curf1cl  18279  curf2cl  18282  curfcl  18283  curfuncf  18289  uncfcurf  18290  diag12  18295  diag2  18296  curf2ndf  18298  hofcl  18310  yon12  18316  yon2  18317  yonedalem3a  18325  yonedalem3b  18330  yonedainv  18332  bj-endmnd  37962  catprs  49789  endmndlem  49793  idmon  49798  idepi  49799  discsubc  49842  imaid  49932  upciclem3  49946  swapfid  50057  tposcurf12  50076  tposcurf2  50078  fucoid  50126  thincid  50210  functhinclem4  50225  termchomn0  50262  idfudiag1  50303  termcarweu  50306  arweutermc  50308