Theorem idaf 16485

Description: The identity arrow function is a function from objects to arrows. (Contributed by Mario Carneiro, 11-Jan-2017.)

Hypotheses

Ref	Expression
idafval.i	⊢ 𝐼 = (Ida‘𝐶)
idafval.b	⊢ 𝐵 = (Base‘𝐶)
idafval.c	⊢ (𝜑 → 𝐶 ∈ Cat)
idaf.a	⊢ 𝐴 = (Arrow‘𝐶)

Assertion

Ref	Expression
idaf	⊢ (𝜑 → 𝐼:𝐵⟶𝐴)

Proof of Theorem idaf

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		otex 4854	. . 3 ⊢ ⟨𝑥, 𝑥, ((Id‘𝐶)‘𝑥)⟩ ∈ V
2	1	a1i 11	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ⟨𝑥, 𝑥, ((Id‘𝐶)‘𝑥)⟩ ∈ V)
3		idafval.i	. . 3 ⊢ 𝐼 = (Ida‘𝐶)
4		idafval.b	. . 3 ⊢ 𝐵 = (Base‘𝐶)
5		idafval.c	. . 3 ⊢ (𝜑 → 𝐶 ∈ Cat)
6		eqid 2610	. . 3 ⊢ (Id‘𝐶) = (Id‘𝐶)
7	3, 4, 5, 6	idafval 16479	. 2 ⊢ (𝜑 → 𝐼 = (𝑥 ∈ 𝐵 ↦ ⟨𝑥, 𝑥, ((Id‘𝐶)‘𝑥)⟩))
8		idaf.a	. . . 4 ⊢ 𝐴 = (Arrow‘𝐶)
9		eqid 2610	. . . 4 ⊢ (Homa‘𝐶) = (Homa‘𝐶)
10	8, 9	homarw 16468	. . 3 ⊢ (𝑥(Homa‘𝐶)𝑥) ⊆ 𝐴
11	5	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝐶 ∈ Cat)
12		simpr 476	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ 𝐵)
13	3, 4, 11, 12, 9	idahom 16482	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝐼‘𝑥) ∈ (𝑥(Homa‘𝐶)𝑥))
14	10, 13	sseldi 3566	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝐼‘𝑥) ∈ 𝐴)
15	2, 7, 14	fmpt2d 6285	1 ⊢ (𝜑 → 𝐼:𝐵⟶𝐴)

Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  Vcvv 3173  ⟨cotp 4133  ⟶wf 5786  ‘cfv 5790  (class class class)co 6527  Basecbs 15644  Catccat 16097  Idccid 16098  Arrowcarw 16444  Hom_achoma 16445  Id_acida 16475

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4694  ax-sep 4704  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6825

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-ot 4134  df-uni 4368  df-iun 4452  df-br 4579  df-opab 4639  df-mpt 4640  df-id 4943  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-riota 6489  df-ov 6530  df-cat 16101  df-cid 16102  df-homa 16448  df-arw 16449  df-ida 16477

This theorem is referenced by: (None)