MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  cidfn Structured version   Visualization version   GIF version

Theorem cidfn 16942
Description: The identity arrow operator is a function from objects to arrows. (Contributed by Mario Carneiro, 4-Jan-2017.)
Hypotheses
Ref Expression
cidfn.b 𝐵 = (Base‘𝐶)
cidfn.i 1 = (Id‘𝐶)
Assertion
Ref Expression
cidfn (𝐶 ∈ Cat → 1 Fn 𝐵)

Proof of Theorem cidfn
Dummy variables 𝑓 𝑔 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 riotaex 7110 . . 3 (𝑔 ∈ (𝑥(Hom ‘𝐶)𝑥)∀𝑦𝐵 (∀𝑓 ∈ (𝑦(Hom ‘𝐶)𝑥)(𝑔(⟨𝑦, 𝑥⟩(comp‘𝐶)𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)(𝑓(⟨𝑥, 𝑥⟩(comp‘𝐶)𝑦)𝑔) = 𝑓)) ∈ V
2 eqid 2819 . . 3 (𝑥𝐵 ↦ (𝑔 ∈ (𝑥(Hom ‘𝐶)𝑥)∀𝑦𝐵 (∀𝑓 ∈ (𝑦(Hom ‘𝐶)𝑥)(𝑔(⟨𝑦, 𝑥⟩(comp‘𝐶)𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)(𝑓(⟨𝑥, 𝑥⟩(comp‘𝐶)𝑦)𝑔) = 𝑓))) = (𝑥𝐵 ↦ (𝑔 ∈ (𝑥(Hom ‘𝐶)𝑥)∀𝑦𝐵 (∀𝑓 ∈ (𝑦(Hom ‘𝐶)𝑥)(𝑔(⟨𝑦, 𝑥⟩(comp‘𝐶)𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)(𝑓(⟨𝑥, 𝑥⟩(comp‘𝐶)𝑦)𝑔) = 𝑓)))
31, 2fnmpti 6484 . 2 (𝑥𝐵 ↦ (𝑔 ∈ (𝑥(Hom ‘𝐶)𝑥)∀𝑦𝐵 (∀𝑓 ∈ (𝑦(Hom ‘𝐶)𝑥)(𝑔(⟨𝑦, 𝑥⟩(comp‘𝐶)𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)(𝑓(⟨𝑥, 𝑥⟩(comp‘𝐶)𝑦)𝑔) = 𝑓))) Fn 𝐵
4 cidfn.b . . . 4 𝐵 = (Base‘𝐶)
5 eqid 2819 . . . 4 (Hom ‘𝐶) = (Hom ‘𝐶)
6 eqid 2819 . . . 4 (comp‘𝐶) = (comp‘𝐶)
7 id 22 . . . 4 (𝐶 ∈ Cat → 𝐶 ∈ Cat)
8 cidfn.i . . . 4 1 = (Id‘𝐶)
94, 5, 6, 7, 8cidfval 16939 . . 3 (𝐶 ∈ Cat → 1 = (𝑥𝐵 ↦ (𝑔 ∈ (𝑥(Hom ‘𝐶)𝑥)∀𝑦𝐵 (∀𝑓 ∈ (𝑦(Hom ‘𝐶)𝑥)(𝑔(⟨𝑦, 𝑥⟩(comp‘𝐶)𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)(𝑓(⟨𝑥, 𝑥⟩(comp‘𝐶)𝑦)𝑔) = 𝑓))))
109fneq1d 6439 . 2 (𝐶 ∈ Cat → ( 1 Fn 𝐵 ↔ (𝑥𝐵 ↦ (𝑔 ∈ (𝑥(Hom ‘𝐶)𝑥)∀𝑦𝐵 (∀𝑓 ∈ (𝑦(Hom ‘𝐶)𝑥)(𝑔(⟨𝑦, 𝑥⟩(comp‘𝐶)𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)(𝑓(⟨𝑥, 𝑥⟩(comp‘𝐶)𝑦)𝑔) = 𝑓))) Fn 𝐵))
113, 10mpbiri 260 1 (𝐶 ∈ Cat → 1 Fn 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 398   = wceq 1530  wcel 2107  wral 3136  cop 4565  cmpt 5137   Fn wfn 6343  cfv 6348  crio 7105  (class class class)co 7148  Basecbs 16475  Hom chom 16568  compcco 16569  Catccat 16927  Idccid 16928
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2791  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-ral 3141  df-rex 3142  df-reu 3143  df-rab 3145  df-v 3495  df-sbc 3771  df-csb 3882  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-sn 4560  df-pr 4562  df-op 4566  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-riota 7106  df-ov 7151  df-cid 16932
This theorem is referenced by:  oppccatid  16981  fucidcl  17227  fucsect  17234  curfcl  17474  curf2ndf  17489
  Copyright terms: Public domain W3C validator