Theorem idaval 18114

Description: Value of the identity arrow function. (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)
idafval.1	⊢ 1 = (Id‘𝐶)
idaval.x	⊢ (𝜑 → 𝑋 ∈ 𝐵)

Assertion

Ref	Expression
idaval	⊢ (𝜑 → (𝐼‘𝑋) = ⟨𝑋, 𝑋, ( 1 ‘𝑋)⟩)

Proof of Theorem idaval

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		idafval.i	. . 3 ⊢ 𝐼 = (Ida‘𝐶)
2		idafval.b	. . 3 ⊢ 𝐵 = (Base‘𝐶)
3		idafval.c	. . 3 ⊢ (𝜑 → 𝐶 ∈ Cat)
4		idafval.1	. . 3 ⊢ 1 = (Id‘𝐶)
5	1, 2, 3, 4	idafval 18113	. 2 ⊢ (𝜑 → 𝐼 = (𝑥 ∈ 𝐵 ↦ ⟨𝑥, 𝑥, ( 1 ‘𝑥)⟩))
6		simpr 489	. . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → 𝑥 = 𝑋)
7	6	fveq2d 6886	. . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → ( 1 ‘𝑥) = ( 1 ‘𝑋))
8	6, 6, 7	oteq123d 4857	. 2 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → ⟨𝑥, 𝑥, ( 1 ‘𝑥)⟩ = ⟨𝑋, 𝑋, ( 1 ‘𝑋)⟩)
9		idaval.x	. 2 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
10		otex 5448	. . 3 ⊢ ⟨𝑋, 𝑋, ( 1 ‘𝑋)⟩ ∈ V
11	10	a1i 11	. 2 ⊢ (𝜑 → ⟨𝑋, 𝑋, ( 1 ‘𝑋)⟩ ∈ V)
12	5, 8, 9, 11	fvmptd 6998	1 ⊢ (𝜑 → (𝐼‘𝑋) = ⟨𝑋, 𝑋, ( 1 ‘𝑋)⟩)

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1567   ∈ wcel 2149  Vcvv 3463  ⟨cotp 4602  ‘cfv 6537  Basecbs 17268  Catccat 17719  Idccid 17720  Id_acida 18109

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5242  ax-sep 5261  ax-nul 5271  ax-pr 5405

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-ot 4603  df-uni 4877  df-iun 4962  df-br 5114  df-opab 5178  df-mpt 5197  df-id 5557  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ida 18111

This theorem is referenced by:  ida2  18115  idahom  18116