Theorem rescbas

Description: Base set of the category restriction. (Contributed by Mario Carneiro, 4-Jan-2017)

	Ref	Expression
Hypotheses	rescbas.d	⊢ 𝐷 = ( 𝐶 ↾cat 𝐻 )
	rescbas.b	⊢ 𝐵 = ( Base ‘ 𝐶 )
	rescbas.c	⊢ ( 𝜑 → 𝐶 ∈ 𝑉 )
	rescbas.h	⊢ ( 𝜑 → 𝐻 Fn ( 𝑆 × 𝑆 ) )
	rescbas.s	⊢ ( 𝜑 → 𝑆 ⊆ 𝐵 )
Assertion	rescbas	⊢ ( 𝜑 → 𝑆 = ( Base ‘ 𝐷 ) )

Proof

Step	Hyp	Ref	Expression
1		rescbas.d	⊢ 𝐷 = ( 𝐶 ↾cat 𝐻 )
2		rescbas.b	⊢ 𝐵 = ( Base ‘ 𝐶 )
3		rescbas.c	⊢ ( 𝜑 → 𝐶 ∈ 𝑉 )
4		rescbas.h	⊢ ( 𝜑 → 𝐻 Fn ( 𝑆 × 𝑆 ) )
5		rescbas.s	⊢ ( 𝜑 → 𝑆 ⊆ 𝐵 )
6		baseid	⊢ Base = Slot ( Base ‘ ndx )
7		1re	⊢ 1 ∈ ℝ
8		1nn	⊢ 1 ∈ ℕ
9		4nn0	⊢ 4 ∈ ℕ0
10		1nn0	⊢ 1 ∈ ℕ0
11		1lt10	⊢ 1 < ; 1 0
12	8 9 10 11	declti	⊢ 1 < ; 1 4
13	7 12	ltneii	⊢ 1 ≠ ; 1 4
14		basendx	⊢ ( Base ‘ ndx ) = 1
15		homndx	⊢ ( Hom ‘ ndx ) = ; 1 4
16	14 15	neeq12i	⊢ ( ( Base ‘ ndx ) ≠ ( Hom ‘ ndx ) ↔ 1 ≠ ; 1 4 )
17	13 16	mpbir	⊢ ( Base ‘ ndx ) ≠ ( Hom ‘ ndx )
18	6 17	setsnid	⊢ ( Base ‘ ( 𝐶 ↾s 𝑆 ) ) = ( Base ‘ ( ( 𝐶 ↾s 𝑆 ) sSet ⟨ ( Hom ‘ ndx ) , 𝐻 ⟩ ) )
19		eqid	⊢ ( 𝐶 ↾s 𝑆 ) = ( 𝐶 ↾s 𝑆 )
20	19 2	ressbas2	⊢ ( 𝑆 ⊆ 𝐵 → 𝑆 = ( Base ‘ ( 𝐶 ↾s 𝑆 ) ) )
21	5 20	syl	⊢ ( 𝜑 → 𝑆 = ( Base ‘ ( 𝐶 ↾s 𝑆 ) ) )
22	2	fvexi	⊢ 𝐵 ∈ V
23	22	ssex	⊢ ( 𝑆 ⊆ 𝐵 → 𝑆 ∈ V )
24	5 23	syl	⊢ ( 𝜑 → 𝑆 ∈ V )
25	1 3 24 4	rescval2	⊢ ( 𝜑 → 𝐷 = ( ( 𝐶 ↾s 𝑆 ) sSet ⟨ ( Hom ‘ ndx ) , 𝐻 ⟩ ) )
26	25	fveq2d	⊢ ( 𝜑 → ( Base ‘ 𝐷 ) = ( Base ‘ ( ( 𝐶 ↾s 𝑆 ) sSet ⟨ ( Hom ‘ ndx ) , 𝐻 ⟩ ) ) )
27	18 21 26	3eqtr4a	⊢ ( 𝜑 → 𝑆 = ( Base ‘ 𝐷 ) )