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Theorem elixp2b 25154
Description: The base class of the elements of a nuple. (Contributed by FL, 6-Jun-2011.) (Revised by Mario Carneiro, 12-Aug-2016.)
Assertion
Ref Expression
elixp2b  |-  ( F  e.  X_ x  e.  A  B  ->  A. x  e.  A  ( F `  x )  e.  B )
Distinct variable group:    x, F
Allowed substitution hints:    A( x)    B( x)

Proof of Theorem elixp2b
Dummy variable  f is distinct from all other variables.
StepHypRef Expression
1 df-ixp 6818 . . . 4  |-  X_ x  e.  A  B  =  { f  |  ( f  Fn  { x  |  x  e.  A }  /\  A. x  e.  A  ( f `  x )  e.  B
) }
2 simpr 447 . . . . 5  |-  ( ( f  Fn  { x  |  x  e.  A }  /\  A. x  e.  A  ( f `  x )  e.  B
)  ->  A. x  e.  A  ( f `  x )  e.  B
)
32ss2abi 3245 . . . 4  |-  { f  |  ( f  Fn 
{ x  |  x  e.  A }  /\  A. x  e.  A  ( f `  x )  e.  B ) } 
C_  { f  | 
A. x  e.  A  ( f `  x
)  e.  B }
41, 3eqsstri 3208 . . 3  |-  X_ x  e.  A  B  C_  { f  |  A. x  e.  A  ( f `  x )  e.  B }
54sseli 3176 . 2  |-  ( F  e.  X_ x  e.  A  B  ->  F  e.  {
f  |  A. x  e.  A  ( f `  x )  e.  B } )
6 fveq1 5524 . . . . 5  |-  ( f  =  F  ->  (
f `  x )  =  ( F `  x ) )
76eleq1d 2349 . . . 4  |-  ( f  =  F  ->  (
( f `  x
)  e.  B  <->  ( F `  x )  e.  B
) )
87ralbidv 2563 . . 3  |-  ( f  =  F  ->  ( A. x  e.  A  ( f `  x
)  e.  B  <->  A. x  e.  A  ( F `  x )  e.  B
) )
98elabg 2915 . 2  |-  ( F  e.  X_ x  e.  A  B  ->  ( F  e. 
{ f  |  A. x  e.  A  (
f `  x )  e.  B }  <->  A. x  e.  A  ( F `  x )  e.  B
) )
105, 9mpbid 201 1  |-  ( F  e.  X_ x  e.  A  B  ->  A. x  e.  A  ( F `  x )  e.  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   {cab 2269   A.wral 2543    Fn wfn 5250   ` cfv 5255   X_cixp 6817
This theorem is referenced by:  bclelnu  25155  dstr  25171
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-rex 2549  df-v 2790  df-in 3159  df-ss 3166  df-uni 3828  df-br 4024  df-iota 5219  df-fv 5263  df-ixp 6818
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