Theorem ixpsnval 6643

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 6642	. 2 |- X_ x e. { X } B = { f | ( f Fn { X } /\ A. x e. { X } ( f ` x ) e. B ) }
2		ralsnsg 3596	. . . . 5 |- ( X e. V -> ( A. x e. { X } ( f ` x ) e. B <-> [. X / x ]. ( f ` x ) e. B ) )
3		sbcel12g 3046	. . . . 5 |- ( X e. V -> ( [. X / x ]. ( f ` x ) e. B <-> [_ X / x ]_ ( f ` x ) e. [_ X / x ]_ B ) )
4		csbfvg 5505	. . . . . 6 |- ( X e. V -> [_ X / x ]_ ( f ` x ) = ( f ` X ) )
5	4	eleq1d 2226	. . . . 5 |- ( X e. V -> ( [_ X / x ]_ ( f ` x ) e. [_ X / x ]_ B <-> ( f ` X ) e. [_ X / x ]_ B ) )
6	2, 3, 5	3bitrd 213	. . . 4 |- ( X e. V -> ( A. x e. { X } ( f ` x ) e. B <-> ( f ` X ) e. [_ X / x ]_ B ) )
7	6	anbi2d 460	. . 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 2275	. 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	syl5eq 2202	1 |- ( X e. V -> X_ x e. { X } B = { f | ( f Fn { X } /\ ( f ` X ) e. [_ X / x ]_ B ) } )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1335   e. wcel 2128   {cab 2143   A.wral 2435   [.wsbc 2937   [_csb 3031   {csn 3560   Fn wfn 5164   ` cfv 5169   X_cixp 6640

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-v 2714  df-sbc 2938  df-csb 3032  df-un 3106  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-br 3966  df-iota 5134  df-fn 5172  df-fv 5177  df-ixp 6641

This theorem is referenced by: (None)