Theorem idaval

Description: Value of the identity arrow function. (Contributed by Mario Carneiro, 11-Jan-2017)

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

Proof

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