Theorem ida2

Description: Morphism part of the identity arrow. (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	ida2	⊢ ( 𝜑 → ( 2nd ‘ ( 𝐼 ‘ 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 5	idaval	⊢ ( 𝜑 → ( 𝐼 ‘ 𝑋 ) = ⟨ 𝑋 , 𝑋 , ( 1 ‘ 𝑋 ) ⟩ )
7	6	fveq2d	⊢ ( 𝜑 → ( 2nd ‘ ( 𝐼 ‘ 𝑋 ) ) = ( 2nd ‘ ⟨ 𝑋 , 𝑋 , ( 1 ‘ 𝑋 ) ⟩ ) )
8		fvex	⊢ ( 1 ‘ 𝑋 ) ∈ V
9		ot3rdg	⊢ ( ( 1 ‘ 𝑋 ) ∈ V → ( 2nd ‘ ⟨ 𝑋 , 𝑋 , ( 1 ‘ 𝑋 ) ⟩ ) = ( 1 ‘ 𝑋 ) )
10	8 9	ax-mp	⊢ ( 2nd ‘ ⟨ 𝑋 , 𝑋 , ( 1 ‘ 𝑋 ) ⟩ ) = ( 1 ‘ 𝑋 )
11	7 10	eqtrdi	⊢ ( 𝜑 → ( 2nd ‘ ( 𝐼 ‘ 𝑋 ) ) = ( 1 ‘ 𝑋 ) )