Theorem setcid

Description: The identity arrow in the category of sets is the identity function. (Contributed by Mario Carneiro, 3-Jan-2017)

	Ref	Expression
Hypotheses	setccat.c	⊢ 𝐶 = ( SetCat ‘ 𝑈 )
	setcid.o	⊢ 1 = ( Id ‘ 𝐶 )
	setcid.u	⊢ ( 𝜑 → 𝑈 ∈ 𝑉 )
	setcid.x	⊢ ( 𝜑 → 𝑋 ∈ 𝑈 )
Assertion	setcid	⊢ ( 𝜑 → ( 1 ‘ 𝑋 ) = ( I ↾ 𝑋 ) )

Proof

Step	Hyp	Ref	Expression
1		setccat.c	⊢ 𝐶 = ( SetCat ‘ 𝑈 )
2		setcid.o	⊢ 1 = ( Id ‘ 𝐶 )
3		setcid.u	⊢ ( 𝜑 → 𝑈 ∈ 𝑉 )
4		setcid.x	⊢ ( 𝜑 → 𝑋 ∈ 𝑈 )
5	1	setccatid	⊢ ( 𝑈 ∈ 𝑉 → ( 𝐶 ∈ Cat ∧ ( Id ‘ 𝐶 ) = ( 𝑥 ∈ 𝑈 ↦ ( I ↾ 𝑥 ) ) ) )
6	3 5	syl	⊢ ( 𝜑 → ( 𝐶 ∈ Cat ∧ ( Id ‘ 𝐶 ) = ( 𝑥 ∈ 𝑈 ↦ ( I ↾ 𝑥 ) ) ) )
7	6	simprd	⊢ ( 𝜑 → ( Id ‘ 𝐶 ) = ( 𝑥 ∈ 𝑈 ↦ ( I ↾ 𝑥 ) ) )
8	2 7	eqtrid	⊢ ( 𝜑 → 1 = ( 𝑥 ∈ 𝑈 ↦ ( I ↾ 𝑥 ) ) )
9		simpr	⊢ ( ( 𝜑 ∧ 𝑥 = 𝑋 ) → 𝑥 = 𝑋 )
10	9	reseq2d	⊢ ( ( 𝜑 ∧ 𝑥 = 𝑋 ) → ( I ↾ 𝑥 ) = ( I ↾ 𝑋 ) )
11	4	resiexd	⊢ ( 𝜑 → ( I ↾ 𝑋 ) ∈ V )
12	8 10 4 11	fvmptd	⊢ ( 𝜑 → ( 1 ‘ 𝑋 ) = ( I ↾ 𝑋 ) )