Theorem ressbas2

Description: Base set of a structure restriction. (Contributed by Mario Carneiro, 2-Dec-2014)

	Ref	Expression
Hypotheses	ressbas.r	⊢ 𝑅 = ( 𝑊 ↾s 𝐴 )
	ressbas.b	⊢ 𝐵 = ( Base ‘ 𝑊 )
Assertion	ressbas2	⊢ ( 𝐴 ⊆ 𝐵 → 𝐴 = ( Base ‘ 𝑅 ) )

Proof

Step	Hyp	Ref	Expression
1		ressbas.r	⊢ 𝑅 = ( 𝑊 ↾s 𝐴 )
2		ressbas.b	⊢ 𝐵 = ( Base ‘ 𝑊 )
3		dfss2	⊢ ( 𝐴 ⊆ 𝐵 ↔ ( 𝐴 ∩ 𝐵 ) = 𝐴 )
4	3	biimpi	⊢ ( 𝐴 ⊆ 𝐵 → ( 𝐴 ∩ 𝐵 ) = 𝐴 )
5	2	fvexi	⊢ 𝐵 ∈ V
6	5	ssex	⊢ ( 𝐴 ⊆ 𝐵 → 𝐴 ∈ V )
7	1 2	ressbas	⊢ ( 𝐴 ∈ V → ( 𝐴 ∩ 𝐵 ) = ( Base ‘ 𝑅 ) )
8	6 7	syl	⊢ ( 𝐴 ⊆ 𝐵 → ( 𝐴 ∩ 𝐵 ) = ( Base ‘ 𝑅 ) )
9	4 8	eqtr3d	⊢ ( 𝐴 ⊆ 𝐵 → 𝐴 = ( Base ‘ 𝑅 ) )