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Theorem ixpsnval 7430
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 7429 . 2  |-  X_ x  e.  { X } B  =  { f  |  ( f  Fn  { X }  /\  A. x  e. 
{ X }  (
f `  x )  e.  B ) }
2 ralsnsg 4003 . . . . 5  |-  ( X  e.  V  ->  ( A. x  e.  { X }  ( f `  x )  e.  B  <->  [. X  /  x ]. ( f `  x
)  e.  B ) )
3 sbcel12 3776 . . . . . 6  |-  ( [. X  /  x ]. (
f `  x )  e.  B  <->  [_ X  /  x ]_ ( f `  x
)  e.  [_ X  /  x ]_ B )
4 csbfv2g 5841 . . . . . . . 8  |-  ( X  e.  V  ->  [_ X  /  x ]_ ( f `
 x )  =  ( f `  [_ X  /  x ]_ x ) )
5 csbvarg 3796 . . . . . . . . 9  |-  ( X  e.  V  ->  [_ X  /  x ]_ x  =  X )
65fveq2d 5809 . . . . . . . 8  |-  ( X  e.  V  ->  (
f `  [_ X  /  x ]_ x )  =  ( f `  X
) )
74, 6eqtrd 2443 . . . . . . 7  |-  ( X  e.  V  ->  [_ X  /  x ]_ ( f `
 x )  =  ( f `  X
) )
87eleq1d 2471 . . . . . 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 702 . . 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 2538 . 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 2455 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 367    = wceq 1405    e. wcel 1842   {cab 2387   A.wral 2753   [.wsbc 3276   [_csb 3372   {csn 3971    Fn wfn 5520   ` cfv 5525   X_cixp 7427
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-nul 4524  ax-pow 4571
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-fal 1411  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-br 4395  df-dm 4952  df-iota 5489  df-fn 5528  df-fv 5533  df-ixp 7428
This theorem is referenced by:  ixpsnbasval  18067
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