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Theorem cidfn 16654
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 6843 . . 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 2799 . . 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 6233 . 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 2799 . . . 4 (Hom ‘𝐶) = (Hom ‘𝐶)
6 eqid 2799 . . . 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 16651 . . 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 6192 . 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 250 1 (𝐶 ∈ Cat → 1 Fn 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 385   = wceq 1653  wcel 2157  wral 3089  cop 4374  cmpt 4922   Fn wfn 6096  cfv 6101  crio 6838  (class class class)co 6878  Basecbs 16184  Hom chom 16278  compcco 16279  Catccat 16639  Idccid 16640
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-rep 4964  ax-sep 4975  ax-nul 4983  ax-pr 5097
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ne 2972  df-ral 3094  df-rex 3095  df-reu 3096  df-rab 3098  df-v 3387  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629  df-iun 4712  df-br 4844  df-opab 4906  df-mpt 4923  df-id 5220  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-rn 5323  df-res 5324  df-ima 5325  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-f1 6106  df-fo 6107  df-f1o 6108  df-fv 6109  df-riota 6839  df-ov 6881  df-cid 16644
This theorem is referenced by:  oppccatid  16693  fucidcl  16939  fucsect  16946  curfcl  17187  curf2ndf  17202
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