Theorem xpsfrn

Description: A short expression for the indexed cartesian product on two indices. (Contributed by Mario Carneiro, 15-Aug-2015)

	Ref	Expression
Hypothesis	xpsff1o.f	|- F = ( x e. A , y e. B |-> { <. (/) , x >. , <. 1o , y >. } )
Assertion	xpsfrn	|- ran F = X_ k e. 2o if ( k = (/) , A , B )

Proof

Step	Hyp	Ref	Expression
1		xpsff1o.f	|- F = ( x e. A , y e. B |-> { <. (/) , x >. , <. 1o , y >. } )
2	1	xpsff1o	|- F : ( A X. B ) -1-1-onto-> X_ k e. 2o if ( k = (/) , A , B )
3		f1ofo	|- ( F : ( A X. B ) -1-1-onto-> X_ k e. 2o if ( k = (/) , A , B ) -> F : ( A X. B ) -onto-> X_ k e. 2o if ( k = (/) , A , B ) )
4		forn	|- ( F : ( A X. B ) -onto-> X_ k e. 2o if ( k = (/) , A , B ) -> ran F = X_ k e. 2o if ( k = (/) , A , B ) )
5	2 3 4	mp2b	|- ran F = X_ k e. 2o if ( k = (/) , A , B )