Theorem xpsff1o2

Description: The function appearing in xpsval is a bijection from the cartesian product to the indexed cartesian product indexed on the pair 2o = { (/) , 1o } . (Contributed by Mario Carneiro, 24-Jan-2015)

	Ref	Expression
Hypothesis	xpsff1o.f	|- F = ( x e. A , y e. B |-> { <. (/) , x >. , <. 1o , y >. } )
Assertion	xpsff1o2	|- F : ( A X. B ) -1-1-onto-> ran F

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		f1of1	|- ( F : ( A X. B ) -1-1-onto-> X_ k e. 2o if ( k = (/) , A , B ) -> F : ( A X. B ) -1-1-> X_ k e. 2o if ( k = (/) , A , B ) )
4		f1f1orn	|- ( F : ( A X. B ) -1-1-> X_ k e. 2o if ( k = (/) , A , B ) -> F : ( A X. B ) -1-1-onto-> ran F )
5	2 3 4	mp2b	|- F : ( A X. B ) -1-1-onto-> ran F