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Theorem elixp2b 25257
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 6834 . . . 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 3258 . . . 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 3221 . . 3  |-  X_ x  e.  A  B  C_  { f  |  A. x  e.  A  ( f `  x )  e.  B }
54sseli 3189 . 2  |-  ( F  e.  X_ x  e.  A  B  ->  F  e.  {
f  |  A. x  e.  A  ( f `  x )  e.  B } )
6 fveq1 5540 . . . . 5  |-  ( f  =  F  ->  (
f `  x )  =  ( F `  x ) )
76eleq1d 2362 . . . 4  |-  ( f  =  F  ->  (
( f `  x
)  e.  B  <->  ( F `  x )  e.  B
) )
87ralbidv 2576 . . 3  |-  ( f  =  F  ->  ( A. x  e.  A  ( f `  x
)  e.  B  <->  A. x  e.  A  ( F `  x )  e.  B
) )
98elabg 2928 . 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 1632    e. wcel 1696   {cab 2282   A.wral 2556    Fn wfn 5266   ` cfv 5271   X_cixp 6833
This theorem is referenced by:  bclelnu  25258  dstr  25274
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ral 2561  df-rex 2562  df-v 2803  df-in 3172  df-ss 3179  df-uni 3844  df-br 4040  df-iota 5235  df-fv 5279  df-ixp 6834
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