Theorem idaval 18020

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 18019	. 2 ⊢ (𝜑 → 𝐼 = (𝑥 ∈ 𝐵 ↦ ⟨𝑥, 𝑥, ( 1 ‘𝑥)⟩))
6		simpr 484	. . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → 𝑥 = 𝑋)
7	6	fveq2d 6862	. . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → ( 1 ‘𝑥) = ( 1 ‘𝑋))
8	6, 6, 7	oteq123d 4852	. 2 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → ⟨𝑥, 𝑥, ( 1 ‘𝑥)⟩ = ⟨𝑋, 𝑋, ( 1 ‘𝑋)⟩)
9		idaval.x	. 2 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
10		otex 5425	. . 3 ⊢ ⟨𝑋, 𝑋, ( 1 ‘𝑋)⟩ ∈ V
11	10	a1i 11	. 2 ⊢ (𝜑 → ⟨𝑋, 𝑋, ( 1 ‘𝑋)⟩ ∈ V)
12	5, 8, 9, 11	fvmptd 6975	1 ⊢ (𝜑 → (𝐼‘𝑋) = ⟨𝑋, 𝑋, ( 1 ‘𝑋)⟩)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  Vcvv 3447  ⟨cotp 4597  ‘cfv 6511  Basecbs 17179  Catccat 17625  Idccid 17626  Id_acida 18015

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5234  ax-sep 5251  ax-nul 5261  ax-pr 5387

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-sn 4590  df-pr 4592  df-op 4596  df-ot 4598  df-uni 4872  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-ida 18017

This theorem is referenced by:  ida2  18021  idahom  18022