Theorem idaval

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

	Ref	Expression
Hypotheses	idafval.i	$$⊢I={Id}_a\left(C\right)$$
	idafval.b	$$⊢B={\mathrm{Base}}_C$$
	idafval.c	$$⊢\phi \to C\in \mathrm{Cat}$$
	idafval.1	$$⊢\underset{\dot{}}{1}=\mathrm{Id}\left(C\right)$$
	idaval.x	$$⊢\phi \to X\in B$$
Assertion	idaval	$$⊢\phi \to I\left(X\right)=⟨X,X,\underset{\dot{}}{1}\left(X\right)⟩$$

Proof

Step	Hyp	Ref	Expression
1		idafval.i	$$⊢I={Id}_a\left(C\right)$$
2		idafval.b	$$⊢B={\mathrm{Base}}_C$$
3		idafval.c	$$⊢\phi \to C\in \mathrm{Cat}$$
4		idafval.1	$$⊢\underset{\dot{}}{1}=\mathrm{Id}\left(C\right)$$
5		idaval.x	$$⊢\phi \to X\in B$$
6	1 2 3 4	idafval	$$⊢\phi \to I=\left(x\in B⟼⟨x,x,\underset{\dot{}}{1}\left(x\right)⟩\right)$$
7		simpr	$$⊢\left(\phi \wedge x=X\right)\to x=X$$
8	7	fveq2d	$$⊢\left(\phi \wedge x=X\right)\to \underset{\dot{}}{1}\left(x\right)=\underset{\dot{}}{1}\left(X\right)$$
9	7 7 8	oteq123d	$$⊢\left(\phi \wedge x=X\right)\to ⟨x,x,\underset{\dot{}}{1}\left(x\right)⟩=⟨X,X,\underset{\dot{}}{1}\left(X\right)⟩$$
10		otex	$$⊢⟨X,X,\underset{\dot{}}{1}\left(X\right)⟩\in \mathrm{V}$$
11	10	a1i	$$⊢\phi \to ⟨X,X,\underset{\dot{}}{1}\left(X\right)⟩\in \mathrm{V}$$
12	6 9 5 11	fvmptd	$$⊢\phi \to I\left(X\right)=⟨X,X,\underset{\dot{}}{1}\left(X\right)⟩$$