ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  ss2ixp Unicode version

Theorem ss2ixp 6612
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 3095 . . . . 5  |-  ( B 
C_  C  ->  (
( f `  x
)  e.  B  -> 
( f `  x
)  e.  C ) )
21ral2imi 2500 . . . 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 335 . . 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 3174 . 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 6600 . 2  |-  X_ x  e.  A  B  =  { f  |  ( f  Fn  { x  |  x  e.  A }  /\  A. x  e.  A  ( f `  x )  e.  B
) }
6 df-ixp 6600 . 2  |-  X_ x  e.  A  C  =  { f  |  ( f  Fn  { x  |  x  e.  A }  /\  A. x  e.  A  ( f `  x )  e.  C
) }
74, 5, 63sstr4g 3144 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 103    e. wcel 1481   {cab 2126   A.wral 2417    C_ wss 3075    Fn wfn 5125   ` cfv 5130   X_cixp 6599
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122
This theorem depends on definitions:  df-bi 116  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-in 3081  df-ss 3088  df-ixp 6600
This theorem is referenced by:  ixpeq2  6613
  Copyright terms: Public domain W3C validator