Theorem catidcl 17596

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 2733	. . 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 17591	. 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 17589	. . 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 7329	. . 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 2833	1 ⊢ (𝜑 → ( 1 ‘𝑋) ∈ (𝑋𝐻𝑋))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2113  ∀wral 3048  ∃!wreu 3345  ⟨cop 4583  ‘cfv 6489  ℩crio 7311  (class class class)co 7355  Basecbs 17127  Hom chom 17179  compcco 17180  Catccat 17578  Idccid 17579

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-rep 5221  ax-sep 5238  ax-nul 5248  ax-pr 5374

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-ral 3049  df-rex 3058  df-rmo 3347  df-reu 3348  df-rab 3397  df-v 3439  df-sbc 3738  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4283  df-if 4477  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4861  df-iun 4945  df-br 5096  df-opab 5158  df-mpt 5177  df-id 5516  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-iota 6445  df-fun 6491  df-fn 6492  df-f 6493  df-f1 6494  df-fo 6495  df-f1o 6496  df-fv 6497  df-riota 7312  df-ov 7358  df-cat 17582  df-cid 17583

This theorem is referenced by:  oppccatid  17633  monsect  17698  sectid  17701  catsubcat  17754  fullsubc  17765  idfucl  17796  cofucl  17803  fthsect  17842  fucidcl  17883  initoid  17916  termoid  17917  idahom  17975  catcisolem  18025  xpccatid  18102  1stfcl  18111  2ndfcl  18112  prfcl  18117  evlfcl  18136  curf1cl  18142  curf2cl  18145  curfcl  18146  curfuncf  18152  uncfcurf  18153  diag12  18158  diag2  18159  curf2ndf  18161  hofcl  18173  yon12  18179  yon2  18180  yonedalem3a  18188  yonedalem3b  18193  yonedainv  18195  bj-endmnd  37435  catprs  49172  endmndlem  49176  idmon  49181  idepi  49182  discsubc  49225  imaid  49315  upciclem3  49329  swapfid  49440  tposcurf12  49459  tposcurf2  49461  fucoid  49509  thincid  49593  functhinclem4  49608  termchomn0  49645  idfudiag1  49686  termcarweu  49689  arweutermc  49691