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 ) )