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Theorem ss2ixp 7268
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 3345 . . . . 5  |-  ( B 
C_  C  ->  (
( f `  x
)  e.  B  -> 
( f `  x
)  e.  C ) )
21ral2imi 2787 . . . 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 565 . . 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 3420 . 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 7256 . 2  |-  X_ x  e.  A  B  =  { f  |  ( f  Fn  { x  |  x  e.  A }  /\  A. x  e.  A  ( f `  x )  e.  B
) }
6 df-ixp 7256 . 2  |-  X_ x  e.  A  C  =  { f  |  ( f  Fn  { x  |  x  e.  A }  /\  A. x  e.  A  ( f `  x )  e.  C
) }
74, 5, 63sstr4g 3392 1  |-  ( A. x  e.  A  B  C_  C  ->  X_ x  e.  A  B  C_  X_ x  e.  A  C )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    e. wcel 1756   {cab 2424   A.wral 2710    C_ wss 3323    Fn wfn 5408   ` cfv 5413   X_cixp 7255
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ral 2715  df-in 3330  df-ss 3337  df-ixp 7256
This theorem is referenced by:  ixpeq2  7269  boxcutc  7298  pwcfsdom  8739  prdsval  14385  prdshom  14397  sscpwex  14720  wunfunc  14801  wunnat  14858  dprdss  16514  psrbaglefi  17418  psrbaglefiOLD  17419  ptuni2  19124  ptcld  19161  ptclsg  19163  prdstopn  19176  xkopt  19203  tmdgsum2  19642  ressprdsds  19921  prdsbl  20041  prdstotbnd  28646
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