Theorem ss2ixp 6811

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 3191	. . . . 5 |- ( B C_ C -> ( ( f ` x ) e. B -> ( f ` x ) e. C ) )
2	1	ral2imi 2572	. . . 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 3270	. 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 6799	. 2 |- X_ x e. A B = { f | ( f Fn { x | x e. A } /\ A. x e. A ( f ` x ) e. B ) }
6		df-ixp 6799	. 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 3240	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 2177   {cab 2192   A.wral 2485   C_ wss 3170   Fn wfn 5275   ` cfv 5280   X_cixp 6798

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2188

This theorem depends on definitions:  df-bi 117  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ral 2490  df-in 3176  df-ss 3183  df-ixp 6799

This theorem is referenced by:  ixpeq2  6812  prdsvallem  13179