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Theorem ss2ixp 7013
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.
StepHypRef Expression
1 ssel 3287 . . . . 5  |-  ( B 
C_  C  ->  (
( f `  x
)  e.  B  -> 
( f `  x
)  e.  C ) )
21ral2imi 2727 . . . 4  |-  ( A. x  e.  A  B  C_  C  ->  ( A. x  e.  A  (
f `  x )  e.  B  ->  A. x  e.  A  ( f `  x )  e.  C
) )
32anim2d 549 . . 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 ) ) )
43ss2abdv 3361 . 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 7002 . 2  |-  X_ x  e.  A  B  =  { f  |  ( f  Fn  { x  |  x  e.  A }  /\  A. x  e.  A  ( f `  x )  e.  B
) }
6 df-ixp 7002 . 2  |-  X_ x  e.  A  C  =  { f  |  ( f  Fn  { x  |  x  e.  A }  /\  A. x  e.  A  ( f `  x )  e.  C
) }
74, 5, 63sstr4g 3334 1  |-  ( A. x  e.  A  B  C_  C  ->  X_ x  e.  A  B  C_  X_ x  e.  A  C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    e. wcel 1717   {cab 2375   A.wral 2651    C_ wss 3265    Fn wfn 5391   ` cfv 5396   X_cixp 7001
This theorem is referenced by:  ixpeq2  7014  boxcutc  7043  pwcfsdom  8393  prdsval  13607  prdshom  13618  sscpwex  13944  wunfunc  14025  wunnat  14082  dprdss  15516  psrbaglefi  16366  ptuni2  17531  ptcld  17568  ptclsg  17570  prdstopn  17583  xkopt  17610  tmdgsum2  18049  ressprdsds  18311  prdsbl  18413  prdstotbnd  26196
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ral 2656  df-in 3272  df-ss 3279  df-ixp 7002
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