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Theorem catidcl 16259
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 (𝜑𝐶 ∈ Cat)
catidcl.x (𝜑𝑋𝐵)
Assertion
Ref Expression
catidcl (𝜑 → ( 1𝑋) ∈ (𝑋𝐻𝑋))

Proof of Theorem catidcl
Dummy variables 𝑓 𝑔 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 catidcl.b . . 3 𝐵 = (Base‘𝐶)
2 catidcl.h . . 3 𝐻 = (Hom ‘𝐶)
3 eqid 2626 . . 3 (comp‘𝐶) = (comp‘𝐶)
4 catidcl.c . . 3 (𝜑𝐶 ∈ Cat)
5 catidcl.i . . 3 1 = (Id‘𝐶)
6 catidcl.x . . 3 (𝜑𝑋𝐵)
71, 2, 3, 4, 5, 6cidval 16254 . 2 (𝜑 → ( 1𝑋) = (𝑔 ∈ (𝑋𝐻𝑋)∀𝑦𝐵 (∀𝑓 ∈ (𝑦𝐻𝑋)(𝑔(⟨𝑦, 𝑋⟩(comp‘𝐶)𝑋)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑋𝐻𝑦)(𝑓(⟨𝑋, 𝑋⟩(comp‘𝐶)𝑦)𝑔) = 𝑓)))
81, 2, 3, 4, 6catideu 16252 . . 3 (𝜑 → ∃!𝑔 ∈ (𝑋𝐻𝑋)∀𝑦𝐵 (∀𝑓 ∈ (𝑦𝐻𝑋)(𝑔(⟨𝑦, 𝑋⟩(comp‘𝐶)𝑋)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑋𝐻𝑦)(𝑓(⟨𝑋, 𝑋⟩(comp‘𝐶)𝑦)𝑔) = 𝑓))
9 riotacl 6580 . . 3 (∃!𝑔 ∈ (𝑋𝐻𝑋)∀𝑦𝐵 (∀𝑓 ∈ (𝑦𝐻𝑋)(𝑔(⟨𝑦, 𝑋⟩(comp‘𝐶)𝑋)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑋𝐻𝑦)(𝑓(⟨𝑋, 𝑋⟩(comp‘𝐶)𝑦)𝑔) = 𝑓) → (𝑔 ∈ (𝑋𝐻𝑋)∀𝑦𝐵 (∀𝑓 ∈ (𝑦𝐻𝑋)(𝑔(⟨𝑦, 𝑋⟩(comp‘𝐶)𝑋)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑋𝐻𝑦)(𝑓(⟨𝑋, 𝑋⟩(comp‘𝐶)𝑦)𝑔) = 𝑓)) ∈ (𝑋𝐻𝑋))
108, 9syl 17 . 2 (𝜑 → (𝑔 ∈ (𝑋𝐻𝑋)∀𝑦𝐵 (∀𝑓 ∈ (𝑦𝐻𝑋)(𝑔(⟨𝑦, 𝑋⟩(comp‘𝐶)𝑋)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑋𝐻𝑦)(𝑓(⟨𝑋, 𝑋⟩(comp‘𝐶)𝑦)𝑔) = 𝑓)) ∈ (𝑋𝐻𝑋))
117, 10eqeltrd 2704 1 (𝜑 → ( 1𝑋) ∈ (𝑋𝐻𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1992  wral 2912  ∃!wreu 2914  cop 4159  cfv 5850  crio 6565  (class class class)co 6605  Basecbs 15776  Hom chom 15868  compcco 15869  Catccat 16241  Idccid 16242
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pr 4872
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3193  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-iota 5813  df-fun 5852  df-fn 5853  df-f 5854  df-f1 5855  df-fo 5856  df-f1o 5857  df-fv 5858  df-riota 6566  df-ov 6608  df-cat 16245  df-cid 16246
This theorem is referenced by:  oppccatid  16295  monsect  16359  sectid  16362  cicref  16377  catsubcat  16415  fullsubc  16426  idfucl  16457  cofucl  16464  fthsect  16501  fucidcl  16541  initoid  16571  termoid  16572  idahom  16626  catcisolem  16672  xpccatid  16744  1stfcl  16753  2ndfcl  16754  prfcl  16759  evlfcl  16778  curf1cl  16784  curf2cl  16787  curfcl  16788  curfuncf  16794  uncfcurf  16795  diag12  16800  diag2  16801  curf2ndf  16803  hofcl  16815  yon12  16821  yon2  16822  yonedalem3a  16830  yonedalem3b  16835  yonedainv  16837
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