Theorem ss2ixp 6923

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 3222	. . . . 5 |- ( B C_ C -> ( ( f ` x ) e. B -> ( f ` x ) e. C ) )
2	1	ral2imi 2598	. . . 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 3301	. 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 6911	. 2 |- X_ x e. A B = { f | ( f Fn { x | x e. A } /\ A. x e. A ( f ` x ) e. B ) }
6		df-ixp 6911	. 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 3271	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 2202   {cab 2217   A.wral 2511   C_ wss 3201   Fn wfn 5328   ` cfv 5333   X_cixp 6910

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-in 3207  df-ss 3214  df-ixp 6911

This theorem is referenced by:  ixpeq2  6924  prdsvallem  13435