Theorem idaval 18112

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 18111	. 2 ⊢ (𝜑 → 𝐼 = (𝑥 ∈ 𝐵 ↦ ⟨𝑥, 𝑥, ( 1 ‘𝑥)⟩))
6		simpr 484	. . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → 𝑥 = 𝑋)
7	6	fveq2d 6911	. . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → ( 1 ‘𝑥) = ( 1 ‘𝑋))
8	6, 6, 7	oteq123d 4893	. 2 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → ⟨𝑥, 𝑥, ( 1 ‘𝑥)⟩ = ⟨𝑋, 𝑋, ( 1 ‘𝑋)⟩)
9		idaval.x	. 2 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
10		otex 5476	. . 3 ⊢ ⟨𝑋, 𝑋, ( 1 ‘𝑋)⟩ ∈ V
11	10	a1i 11	. 2 ⊢ (𝜑 → ⟨𝑋, 𝑋, ( 1 ‘𝑋)⟩ ∈ V)
12	5, 8, 9, 11	fvmptd 7023	1 ⊢ (𝜑 → (𝐼‘𝑋) = ⟨𝑋, 𝑋, ( 1 ‘𝑋)⟩)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1537   ∈ wcel 2106  Vcvv 3478  ⟨cotp 4639  ‘cfv 6563  Basecbs 17245  Catccat 17709  Idccid 17710  Id_acida 18107

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pr 5438

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-ot 4640  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ida 18109

This theorem is referenced by:  ida2  18113  idahom  18114