Theorem ixpsnval 6788

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 6787	. 2 |- X_ x e. { X } B = { f | ( f Fn { X } /\ A. x e. { X } ( f ` x ) e. B ) }
2		ralsnsg 3670	. . . . 5 |- ( X e. V -> ( A. x e. { X } ( f ` x ) e. B <-> [. X / x ]. ( f ` x ) e. B ) )
3		sbcel12g 3108	. . . . 5 |- ( X e. V -> ( [. X / x ]. ( f ` x ) e. B <-> [_ X / x ]_ ( f ` x ) e. [_ X / x ]_ B ) )
4		csbfvg 5616	. . . . . 6 |- ( X e. V -> [_ X / x ]_ ( f ` x ) = ( f ` X ) )
5	4	eleq1d 2274	. . . . 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 2323	. 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 2250	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 1373   e. wcel 2176   {cab 2191   A.wral 2484   [.wsbc 2998   [_csb 3093   {csn 3633   Fn wfn 5266   ` cfv 5271   X_cixp 6785

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rex 2490  df-v 2774  df-sbc 2999  df-csb 3094  df-un 3170  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-br 4045  df-iota 5232  df-fn 5274  df-fv 5279  df-ixp 6786

This theorem is referenced by:  ixpsnbasval  14228