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Theorem ixpsnval 6643
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 6642 . 2  |-  X_ x  e.  { X } B  =  { f  |  ( f  Fn  { X }  /\  A. x  e. 
{ X }  (
f `  x )  e.  B ) }
2 ralsnsg 3596 . . . . 5  |-  ( X  e.  V  ->  ( A. x  e.  { X }  ( f `  x )  e.  B  <->  [. X  /  x ]. ( f `  x
)  e.  B ) )
3 sbcel12g 3046 . . . . 5  |-  ( X  e.  V  ->  ( [. X  /  x ]. ( f `  x
)  e.  B  <->  [_ X  /  x ]_ ( f `  x )  e.  [_ X  /  x ]_ B
) )
4 csbfvg 5505 . . . . . 6  |-  ( X  e.  V  ->  [_ X  /  x ]_ ( f `
 x )  =  ( f `  X
) )
54eleq1d 2226 . . . . 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 460 . . 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 2275 . 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 2202 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 1335    e. wcel 2128   {cab 2143   A.wral 2435   [.wsbc 2937   [_csb 3031   {csn 3560    Fn wfn 5164   ` cfv 5169   X_cixp 6640
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 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-v 2714  df-sbc 2938  df-csb 3032  df-un 3106  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-br 3966  df-iota 5134  df-fn 5172  df-fv 5177  df-ixp 6641
This theorem is referenced by: (None)
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