Theorem ixpsnval 6936

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

Step	Hyp	Ref	Expression
1		dfixp 6935	. 2 |- X_ x e. { X } B = { f | ( f Fn { X } /\ A. x e. { X } ( f ` x ) e. B ) }
2		ralsnsg 3726	. . . . 5 |- ( X e. V -> ( A. x e. { X } ( f ` x ) e. B <-> [. X / x ]. ( f ` x ) e. B ) )
3		sbcel12g 3153	. . . . 5 |- ( X e. V -> ( [. X / x ]. ( f ` x ) e. B <-> [_ X / x ]_ ( f ` x ) e. [_ X / x ]_ B ) )
4		csbfvg 5712	. . . . . 6 |- ( X e. V -> [_ X / x ]_ ( f ` x ) = ( f ` X ) )
5	4	eleq1d 2301	. . . . 5 |- ( X e. V -> ( [_ X / x ]_ ( f ` x ) e. [_ X / x ]_ B <-> ( f ` X ) e. [_ X / x ]_ B ) )
6	2, 3, 5	3bitrd 214	. . . 4 |- ( X e. V -> ( A. x e. { X } ( f ` x ) e. B <-> ( f ` X ) e. [_ X / x ]_ B ) )
7	6	anbi2d 464	. . 3 |- ( X e. V -> ( ( f Fn { X } /\ A. x e. { X } ( f ` x ) e. B ) <-> ( f Fn { X } /\ ( f ` X ) e. [_ X / x ]_ B ) ) )
8	7	abbidv 2352	. 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 ) } )
9	1, 8	eqtrid 2277	1 |- ( X e. V -> X_ x e. { X } B = { f | ( f Fn { X } /\ ( f ` X ) e. [_ X / x ]_ B ) } )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1398   e. wcel 2203   {cab 2218   A.wral 2520   [.wsbc 3042   [_csb 3138   {csn 3689   Fn wfn 5347   ` cfv 5352   X_cixp 6933

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2214

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-v 2815  df-sbc 3043  df-csb 3139  df-un 3215  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-br 4110  df-iota 5312  df-fn 5355  df-fv 5360  df-ixp 6934

This theorem is referenced by:  ixpsnbasval  14614