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Theorem ixpsnval 6679
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 6678 . 2  |-  X_ x  e.  { X } B  =  { f  |  ( f  Fn  { X }  /\  A. x  e. 
{ X }  (
f `  x )  e.  B ) }
2 ralsnsg 3620 . . . . 5  |-  ( X  e.  V  ->  ( A. x  e.  { X }  ( f `  x )  e.  B  <->  [. X  /  x ]. ( f `  x
)  e.  B ) )
3 sbcel12g 3064 . . . . 5  |-  ( X  e.  V  ->  ( [. X  /  x ]. ( f `  x
)  e.  B  <->  [_ X  /  x ]_ ( f `  x )  e.  [_ X  /  x ]_ B
) )
4 csbfvg 5534 . . . . . 6  |-  ( X  e.  V  ->  [_ X  /  x ]_ ( f `
 x )  =  ( f `  X
) )
54eleq1d 2239 . . . . 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 461 . . 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 2288 . 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, 8eqtrid 2215 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 1348    e. wcel 2141   {cab 2156   A.wral 2448   [.wsbc 2955   [_csb 3049   {csn 3583    Fn wfn 5193   ` cfv 5198   X_cixp 6676
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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-sbc 2956  df-csb 3050  df-un 3125  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-iota 5160  df-fn 5201  df-fv 5206  df-ixp 6677
This theorem is referenced by: (None)
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