Theorem restin

Description: When the subspace region is not a subset of the base of the topology, the resulting set is the same as the subspace restricted to the base. (Contributed by Mario Carneiro, 15-Dec-2013)

	Ref	Expression
Hypothesis	restin.1	|- X = U. J
Assertion	restin	|- ( ( J e. V /\ A e. W ) -> ( J |`t A ) = ( J |`t ( A i^i X ) ) )

Proof

Step	Hyp	Ref	Expression
1		restin.1	|- X = U. J
2		uniexg	|- ( J e. V -> U. J e. _V )
3	1 2	eqeltrid	|- ( J e. V -> X e. _V )
4	3	adantr	|- ( ( J e. V /\ A e. W ) -> X e. _V )
5		restco	|- ( ( J e. V /\ X e. _V /\ A e. W ) -> ( ( J |`t X ) |`t A ) = ( J |`t ( X i^i A ) ) )
6	5	3com23	|- ( ( J e. V /\ A e. W /\ X e. _V ) -> ( ( J |`t X ) |`t A ) = ( J |`t ( X i^i A ) ) )
7	4 6	mpd3an3	|- ( ( J e. V /\ A e. W ) -> ( ( J |`t X ) |`t A ) = ( J |`t ( X i^i A ) ) )
8	1	restid	|- ( J e. V -> ( J |`t X ) = J )
9	8	adantr	|- ( ( J e. V /\ A e. W ) -> ( J |`t X ) = J )
10	9	oveq1d	|- ( ( J e. V /\ A e. W ) -> ( ( J |`t X ) |`t A ) = ( J |`t A ) )
11		incom	|- ( X i^i A ) = ( A i^i X )
12	11	oveq2i	|- ( J |`t ( X i^i A ) ) = ( J |`t ( A i^i X ) )
13	12	a1i	|- ( ( J e. V /\ A e. W ) -> ( J |`t ( X i^i A ) ) = ( J |`t ( A i^i X ) ) )
14	7 10 13	3eqtr3d	|- ( ( J e. V /\ A e. W ) -> ( J |`t A ) = ( J |`t ( A i^i X ) ) )