Theorem ss2ixp 6770

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 3177	. . . . 5 |- ( B C_ C -> ( ( f ` x ) e. B -> ( f ` x ) e. C ) )
2	1	ral2imi 2562	. . . 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 337	. . 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 3256	. 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 6758	. 2 |- X_ x e. A B = { f | ( f Fn { x | x e. A } /\ A. x e. A ( f ` x ) e. B ) }
6		df-ixp 6758	. 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 3226	1 |- ( A. x e. A B C_ C -> X_ x e. A B C_ X_ x e. A C )

Syntax hints:   -> wi 4   /\ wa 104   e. wcel 2167   {cab 2182   A.wral 2475   C_ wss 3157   Fn wfn 5253   ` cfv 5258   X_cixp 6757

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-in 3163  df-ss 3170  df-ixp 6758

This theorem is referenced by:  ixpeq2  6771