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Metamath Proof Explorer

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 ) ) )