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Theorem ixpsnval 6588
Description: The value of an infinite Cartesian product with a singleton. (Contributed by AV, 3-Dec-2018.)
Assertion
Ref Expression
ixpsnval  |-  ( X  e.  V  ->  X_ x  e.  { X } B  =  { f  |  ( f  Fn  { X }  /\  ( f `  X )  e.  [_ X  /  x ]_ B
) } )
Distinct variable groups:    B, f    f, V    f, X, x
Allowed substitution hints:    B( x)    V( x)

Proof of Theorem ixpsnval
StepHypRef Expression
1 dfixp 6587 . 2  |-  X_ x  e.  { X } B  =  { f  |  ( f  Fn  { X }  /\  A. x  e. 
{ X }  (
f `  x )  e.  B ) }
2 ralsnsg 3556 . . . . 5  |-  ( X  e.  V  ->  ( A. x  e.  { X }  ( f `  x )  e.  B  <->  [. X  /  x ]. ( f `  x
)  e.  B ) )
3 sbcel12g 3012 . . . . 5  |-  ( X  e.  V  ->  ( [. X  /  x ]. ( f `  x
)  e.  B  <->  [_ X  /  x ]_ ( f `  x )  e.  [_ X  /  x ]_ B
) )
4 csbfvg 5452 . . . . . 6  |-  ( X  e.  V  ->  [_ X  /  x ]_ ( f `
 x )  =  ( f `  X
) )
54eleq1d 2206 . . . . 5  |-  ( X  e.  V  ->  ( [_ X  /  x ]_ ( f `  x
)  e.  [_ X  /  x ]_ B  <->  ( f `  X )  e.  [_ X  /  x ]_ B
) )
62, 3, 53bitrd 213 . . . 4  |-  ( X  e.  V  ->  ( A. x  e.  { X }  ( f `  x )  e.  B  <->  ( f `  X )  e.  [_ X  /  x ]_ B ) )
76anbi2d 459 . . 3  |-  ( X  e.  V  ->  (
( f  Fn  { X }  /\  A. x  e.  { X }  (
f `  x )  e.  B )  <->  ( f  Fn  { X }  /\  ( f `  X
)  e.  [_ X  /  x ]_ B ) ) )
87abbidv 2255 . 2  |-  ( X  e.  V  ->  { f  |  ( f  Fn 
{ X }  /\  A. x  e.  { X }  ( f `  x )  e.  B
) }  =  {
f  |  ( f  Fn  { X }  /\  ( f `  X
)  e.  [_ X  /  x ]_ B ) } )
91, 8syl5eq 2182 1  |-  ( X  e.  V  ->  X_ x  e.  { X } B  =  { f  |  ( f  Fn  { X }  /\  ( f `  X )  e.  [_ X  /  x ]_ B
) } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1331    e. wcel 1480   {cab 2123   A.wral 2414   [.wsbc 2904   [_csb 2998   {csn 3522    Fn wfn 5113   ` cfv 5118   X_cixp 6585
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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-v 2683  df-sbc 2905  df-csb 2999  df-un 3070  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-br 3925  df-iota 5083  df-fn 5121  df-fv 5126  df-ixp 6586
This theorem is referenced by: (None)
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