Theorem ss2ixp 6605

Description: Subclass theorem for infinite Cartesian product. (Contributed by NM, 29-Sep-2006.) (Revised by Mario Carneiro, 12-Aug-2016.)

Assertion

Ref	Expression
ss2ixp	|- ( A. x e. A B C_ C -> X_ x e. A B C_ X_ x e. A C )

Proof of Theorem ss2ixp

Dummy variable f is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ssel 3091	. . . . 5 |- ( B C_ C -> ( ( f ` x ) e. B -> ( f ` x ) e. C ) )
2	1	ral2imi 2497	. . . 4 |- ( A. x e. A B C_ C -> ( A. x e. A ( f ` x ) e. B -> A. x e. A ( f ` x ) e. C ) )
3	2	anim2d 335	. . 3 |- ( A. x e. A B C_ C -> ( ( f Fn { x | x e. A } /\ A. x e. A ( f ` x ) e. B ) -> ( f Fn { x | x e. A } /\ A. x e. A ( f ` x ) e. C ) ) )
4	3	ss2abdv 3170	. 2 |- ( A. x e. A B C_ C -> { f | ( f Fn { x | x e. A } /\ A. x e. A ( f ` x ) e. B ) } C_ { f | ( f Fn { x | x e. A } /\ A. x e. A ( f ` x ) e. C ) } )
5		df-ixp 6593	. 2 |- X_ x e. A B = { f | ( f Fn { x | x e. A } /\ A. x e. A ( f ` x ) e. B ) }
6		df-ixp 6593	. 2 |- X_ x e. A C = { f | ( f Fn { x | x e. A } /\ A. x e. A ( f ` x ) e. C ) }
7	4, 5, 6	3sstr4g 3140	1 |- ( A. x e. A B C_ C -> X_ x e. A B C_ X_ x e. A C )

Syntax hints:   -> wi 4   /\ wa 103   e. wcel 1480   {cab 2125   A.wral 2416   C_ wss 3071   Fn wfn 5118   ` cfv 5123   X_cixp 6592

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-in 3077  df-ss 3084  df-ixp 6593

This theorem is referenced by:  ixpeq2  6606