Theorem idaval 18075

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 18074	. 2 ⊢ (𝜑 → 𝐼 = (𝑥 ∈ 𝐵 ↦ ⟨𝑥, 𝑥, ( 1 ‘𝑥)⟩))
6		simpr 483	. . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → 𝑥 = 𝑋)
7	6	fveq2d 6897	. . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → ( 1 ‘𝑥) = ( 1 ‘𝑋))
8	6, 6, 7	oteq123d 4886	. 2 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → ⟨𝑥, 𝑥, ( 1 ‘𝑥)⟩ = ⟨𝑋, 𝑋, ( 1 ‘𝑋)⟩)
9		idaval.x	. 2 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
10		otex 5463	. . 3 ⊢ ⟨𝑋, 𝑋, ( 1 ‘𝑋)⟩ ∈ V
11	10	a1i 11	. 2 ⊢ (𝜑 → ⟨𝑋, 𝑋, ( 1 ‘𝑋)⟩ ∈ V)
12	5, 8, 9, 11	fvmptd 7008	1 ⊢ (𝜑 → (𝐼‘𝑋) = ⟨𝑋, 𝑋, ( 1 ‘𝑋)⟩)

Syntax hints:   → wi 4   ∧ wa 394   = wceq 1534   ∈ wcel 2099  Vcvv 3462  ⟨cotp 4631  ‘cfv 6546  Basecbs 17208  Catccat 17672  Idccid 17673  Id_acida 18070

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-rep 5282  ax-sep 5296  ax-nul 5303  ax-pr 5425

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3464  df-sbc 3776  df-csb 3892  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-nul 4323  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-ot 4632  df-uni 4906  df-iun 4995  df-br 5146  df-opab 5208  df-mpt 5229  df-id 5572  df-xp 5680  df-rel 5681  df-cnv 5682  df-co 5683  df-dm 5684  df-rn 5685  df-res 5686  df-ima 5687  df-iota 6498  df-fun 6548  df-fn 6549  df-f 6550  df-f1 6551  df-fo 6552  df-f1o 6553  df-fv 6554  df-ida 18072

This theorem is referenced by:  ida2  18076  idahom  18077