Theorem cidfn 17723

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.

Step	Hyp	Ref	Expression
1		riotaex 7393	. . 3 ⊢ (℩𝑔 ∈ (𝑥(Hom ‘𝐶)𝑥)∀𝑦 ∈ 𝐵 (∀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 2736	. . 3 ⊢ (𝑥 ∈ 𝐵 ↦ (℩𝑔 ∈ (𝑥(Hom ‘𝐶)𝑥)∀𝑦 ∈ 𝐵 (∀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))) = (𝑥 ∈ 𝐵 ↦ (℩𝑔 ∈ (𝑥(Hom ‘𝐶)𝑥)∀𝑦 ∈ 𝐵 (∀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)))
3	1, 2	fnmpti 6710	. 2 ⊢ (𝑥 ∈ 𝐵 ↦ (℩𝑔 ∈ (𝑥(Hom ‘𝐶)𝑥)∀𝑦 ∈ 𝐵 (∀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 2736	. . . 4 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
6		eqid 2736	. . . 4 ⊢ (comp‘𝐶) = (comp‘𝐶)
7		id 22	. . . 4 ⊢ (𝐶 ∈ Cat → 𝐶 ∈ Cat)
8		cidfn.i	. . . 4 ⊢ 1 = (Id‘𝐶)
9	4, 5, 6, 7, 8	cidfval 17720	. . 3 ⊢ (𝐶 ∈ Cat → 1 = (𝑥 ∈ 𝐵 ↦ (℩𝑔 ∈ (𝑥(Hom ‘𝐶)𝑥)∀𝑦 ∈ 𝐵 (∀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))))
10	9	fneq1d 6660	. 2 ⊢ (𝐶 ∈ Cat → ( 1 Fn 𝐵 ↔ (𝑥 ∈ 𝐵 ↦ (℩𝑔 ∈ (𝑥(Hom ‘𝐶)𝑥)∀𝑦 ∈ 𝐵 (∀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))
11	3, 10	mpbiri 258	1 ⊢ (𝐶 ∈ Cat → 1 Fn 𝐵)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1539   ∈ wcel 2107  ∀wral 3060  ⟨cop 4631   ↦ cmpt 5224   Fn wfn 6555  ‘cfv 6560  ℩crio 7388  (class class class)co 7432  Basecbs 17248  Hom chom 17309  compcco 17310  Catccat 17708  Idccid 17709

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pr 5431

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-riota 7389  df-ov 7435  df-cid 17713

This theorem is referenced by:  oppccatid  17763  fucidcl  18014  fucsect  18021  curfcl  18278  curf2ndf  18293  fucoid  49066