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Theorem ixpsnval 7469
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 7468 . 2  |-  X_ x  e.  { X } B  =  { f  |  ( f  Fn  { X }  /\  A. x  e. 
{ X }  (
f `  x )  e.  B ) }
2 ralsnsg 4059 . . . . 5  |-  ( X  e.  V  ->  ( A. x  e.  { X }  ( f `  x )  e.  B  <->  [. X  /  x ]. ( f `  x
)  e.  B ) )
3 sbcel12 3823 . . . . . 6  |-  ( [. X  /  x ]. (
f `  x )  e.  B  <->  [_ X  /  x ]_ ( f `  x
)  e.  [_ X  /  x ]_ B )
4 csbfv2g 5901 . . . . . . . 8  |-  ( X  e.  V  ->  [_ X  /  x ]_ ( f `
 x )  =  ( f `  [_ X  /  x ]_ x ) )
5 csbvarg 3848 . . . . . . . . 9  |-  ( X  e.  V  ->  [_ X  /  x ]_ x  =  X )
65fveq2d 5868 . . . . . . . 8  |-  ( X  e.  V  ->  (
f `  [_ X  /  x ]_ x )  =  ( f `  X
) )
74, 6eqtrd 2508 . . . . . . 7  |-  ( X  e.  V  ->  [_ X  /  x ]_ ( f `
 x )  =  ( f `  X
) )
87eleq1d 2536 . . . . . 6  |-  ( X  e.  V  ->  ( [_ X  /  x ]_ ( f `  x
)  e.  [_ X  /  x ]_ B  <->  ( f `  X )  e.  [_ X  /  x ]_ B
) )
93, 8syl5bb 257 . . . . 5  |-  ( X  e.  V  ->  ( [. X  /  x ]. ( f `  x
)  e.  B  <->  ( f `  X )  e.  [_ X  /  x ]_ B
) )
102, 9bitrd 253 . . . 4  |-  ( X  e.  V  ->  ( A. x  e.  { X }  ( f `  x )  e.  B  <->  ( f `  X )  e.  [_ X  /  x ]_ B ) )
1110anbi2d 703 . . 3  |-  ( X  e.  V  ->  (
( f  Fn  { X }  /\  A. x  e.  { X }  (
f `  x )  e.  B )  <->  ( f  Fn  { X }  /\  ( f `  X
)  e.  [_ X  /  x ]_ B ) ) )
1211abbidv 2603 . 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 ) } )
131, 12syl5eq 2520 1  |-  ( X  e.  V  ->  X_ x  e.  { X } B  =  { f  |  ( f  Fn  { X }  /\  ( f `  X )  e.  [_ X  /  x ]_ B
) } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767   {cab 2452   A.wral 2814   [.wsbc 3331   [_csb 3435   {csn 4027    Fn wfn 5581   ` cfv 5586   X_cixp 7466
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-nul 4576  ax-pow 4625
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-dm 5009  df-iota 5549  df-fn 5589  df-fv 5594  df-ixp 7467
This theorem is referenced by:  ixpsnbasval  17638
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