Theorem ixpsnval 6846

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 6845	. 2 |- X_ x e. { X } B = { f | ( f Fn { X } /\ A. x e. { X } ( f ` x ) e. B ) }
2		ralsnsg 3703	. . . . 5 |- ( X e. V -> ( A. x e. { X } ( f ` x ) e. B <-> [. X / x ]. ( f ` x ) e. B ) )
3		sbcel12g 3139	. . . . 5 |- ( X e. V -> ( [. X / x ]. ( f ` x ) e. B <-> [_ X / x ]_ ( f ` x ) e. [_ X / x ]_ B ) )
4		csbfvg 5668	. . . . . 6 |- ( X e. V -> [_ X / x ]_ ( f ` x ) = ( f ` X ) )
5	4	eleq1d 2298	. . . . 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 2347	. 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 2274	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 1395   e. wcel 2200   {cab 2215   A.wral 2508   [.wsbc 3028   [_csb 3124   {csn 3666   Fn wfn 5312   ` cfv 5317   X_cixp 6843

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-sbc 3029  df-csb 3125  df-un 3201  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-br 4083  df-iota 5277  df-fn 5320  df-fv 5325  df-ixp 6844

This theorem is referenced by:  ixpsnbasval  14424