Theorem xpsdsfn2

Description: Closure of the metric in a binary structure product. (Contributed by Mario Carneiro, 20-Aug-2015)

	Ref	Expression
Hypotheses	xpsds.t	|- T = ( R Xs. S )
	xpsds.x	|- X = ( Base ` R )
	xpsds.y	|- Y = ( Base ` S )
	xpsds.1	|- ( ph -> R e. V )
	xpsds.2	|- ( ph -> S e. W )
	xpsds.p	|- P = ( dist ` T )
Assertion	xpsdsfn2	|- ( ph -> P Fn ( ( Base ` T ) X. ( Base ` T ) ) )

Proof

Step	Hyp	Ref	Expression
1		xpsds.t	|- T = ( R Xs. S )
2		xpsds.x	|- X = ( Base ` R )
3		xpsds.y	|- Y = ( Base ` S )
4		xpsds.1	|- ( ph -> R e. V )
5		xpsds.2	|- ( ph -> S e. W )
6		xpsds.p	|- P = ( dist ` T )
7	1 2 3 4 5 6	xpsdsfn	|- ( ph -> P Fn ( ( X X. Y ) X. ( X X. Y ) ) )
8	1 2 3 4 5	xpsbas	|- ( ph -> ( X X. Y ) = ( Base ` T ) )
9	8	sqxpeqd	|- ( ph -> ( ( X X. Y ) X. ( X X. Y ) ) = ( ( Base ` T ) X. ( Base ` T ) ) )
10	9	fneq2d	|- ( ph -> ( P Fn ( ( X X. Y ) X. ( X X. Y ) ) <-> P Fn ( ( Base ` T ) X. ( Base ` T ) ) ) )
11	7 10	mpbid	|- ( ph -> P Fn ( ( Base ` T ) X. ( Base ` T ) ) )