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Metamath Proof Explorer

Theorem xpsbas

Description: The base set of the binary structure product. (Contributed by Mario Carneiro, 15-Aug-2015)

Ref Expression
Hypotheses xpsval.t 
|- T = ( R Xs. S )
xpsval.x 
|- X = ( Base ` R )
xpsval.y 
|- Y = ( Base ` S )
xpsval.1 
|- ( ph -> R e. V )
xpsval.2 
|- ( ph -> S e. W )
Assertion xpsbas 
|- ( ph -> ( X X. Y ) = ( Base ` T ) )

Proof

Step Hyp Ref Expression
1 xpsval.t 
 |-  T = ( R Xs. S )
2 xpsval.x 
 |-  X = ( Base ` R )
3 xpsval.y 
 |-  Y = ( Base ` S )
4 xpsval.1 
 |-  ( ph -> R e. V )
5 xpsval.2 
 |-  ( ph -> S e. W )
6 eqid 
 |-  ( x e. X , y e. Y |-> { <. (/) , x >. , <. 1o , y >. } ) = ( x e. X , y e. Y |-> { <. (/) , x >. , <. 1o , y >. } )
7 eqid 
 |-  ( Scalar ` R ) = ( Scalar ` R )
8 eqid 
 |-  ( ( Scalar ` R ) Xs_ { <. (/) , R >. , <. 1o , S >. } ) = ( ( Scalar ` R ) Xs_ { <. (/) , R >. , <. 1o , S >. } )
9 1 2 3 4 5 6 7 8 xpsval 
 |-  ( ph -> T = ( `' ( x e. X , y e. Y |-> { <. (/) , x >. , <. 1o , y >. } ) "s ( ( Scalar ` R ) Xs_ { <. (/) , R >. , <. 1o , S >. } ) ) )
10 1 2 3 4 5 6 7 8 xpsrnbas 
 |-  ( ph -> ran ( x e. X , y e. Y |-> { <. (/) , x >. , <. 1o , y >. } ) = ( Base ` ( ( Scalar ` R ) Xs_ { <. (/) , R >. , <. 1o , S >. } ) ) )
11 6 xpsff1o2 
 |-  ( x e. X , y e. Y |-> { <. (/) , x >. , <. 1o , y >. } ) : ( X X. Y ) -1-1-onto-> ran ( x e. X , y e. Y |-> { <. (/) , x >. , <. 1o , y >. } )
12 f1ocnv 
 |-  ( ( x e. X , y e. Y |-> { <. (/) , x >. , <. 1o , y >. } ) : ( X X. Y ) -1-1-onto-> ran ( x e. X , y e. Y |-> { <. (/) , x >. , <. 1o , y >. } ) -> `' ( x e. X , y e. Y |-> { <. (/) , x >. , <. 1o , y >. } ) : ran ( x e. X , y e. Y |-> { <. (/) , x >. , <. 1o , y >. } ) -1-1-onto-> ( X X. Y ) )
13 11 12 ax-mp 
 |-  `' ( x e. X , y e. Y |-> { <. (/) , x >. , <. 1o , y >. } ) : ran ( x e. X , y e. Y |-> { <. (/) , x >. , <. 1o , y >. } ) -1-1-onto-> ( X X. Y )
14 f1ofo 
 |-  ( `' ( x e. X , y e. Y |-> { <. (/) , x >. , <. 1o , y >. } ) : ran ( x e. X , y e. Y |-> { <. (/) , x >. , <. 1o , y >. } ) -1-1-onto-> ( X X. Y ) -> `' ( x e. X , y e. Y |-> { <. (/) , x >. , <. 1o , y >. } ) : ran ( x e. X , y e. Y |-> { <. (/) , x >. , <. 1o , y >. } ) -onto-> ( X X. Y ) )
15 13 14 mp1i 
 |-  ( ph -> `' ( x e. X , y e. Y |-> { <. (/) , x >. , <. 1o , y >. } ) : ran ( x e. X , y e. Y |-> { <. (/) , x >. , <. 1o , y >. } ) -onto-> ( X X. Y ) )
16 ovexd 
 |-  ( ph -> ( ( Scalar ` R ) Xs_ { <. (/) , R >. , <. 1o , S >. } ) e. _V )
17 9 10 15 16 imasbas 
 |-  ( ph -> ( X X. Y ) = ( Base ` T ) )