Theorem ixpsnval 7911

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 7910	. 2 |- X_ x e. { X } B = { f | ( f Fn { X } /\ A. x e. { X } ( f ` x ) e. B ) }
2		ralsnsg 4216	. . . . 5 |- ( X e. V -> ( A. x e. { X } ( f ` x ) e. B <-> [. X / x ]. ( f ` x ) e. B ) )
3		sbcel12 3983	. . . . . 6 |- ( [. X / x ]. ( f ` x ) e. B <-> [_ X / x ]_ ( f ` x ) e. [_ X / x ]_ B )
4		csbfv2g 6232	. . . . . . . 8 |- ( X e. V -> [_ X / x ]_ ( f ` x ) = ( f ` [_ X / x ]_ x ) )
5		csbvarg 4003	. . . . . . . . 9 |- ( X e. V -> [_ X / x ]_ x = X )
6	5	fveq2d 6195	. . . . . . . 8 |- ( X e. V -> ( f ` [_ X / x ]_ x ) = ( f ` X ) )
7	4, 6	eqtrd 2656	. . . . . . 7 |- ( X e. V -> [_ X / x ]_ ( f ` x ) = ( f ` X ) )
8	7	eleq1d 2686	. . . . . 6 |- ( X e. V -> ( [_ X / x ]_ ( f ` x ) e. [_ X / x ]_ B <-> ( f ` X ) e. [_ X / x ]_ B ) )
9	3, 8	syl5bb 272	. . . . 5 |- ( X e. V -> ( [. X / x ]. ( f ` x ) e. B <-> ( f ` X ) e. [_ X / x ]_ B ) )
10	2, 9	bitrd 268	. . . 4 |- ( X e. V -> ( A. x e. { X } ( f ` x ) e. B <-> ( f ` X ) e. [_ X / x ]_ B ) )
11	10	anbi2d 740	. . 3 |- ( X e. V -> ( ( f Fn { X } /\ A. x e. { X } ( f ` x ) e. B ) <-> ( f Fn { X } /\ ( f ` X ) e. [_ X / x ]_ B ) ) )
12	11	abbidv 2741	. 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 ) } )
13	1, 12	syl5eq 2668	1 |- ( X e. V -> X_ x e. { X } B = { f | ( f Fn { X } /\ ( f ` X ) e. [_ X / x ]_ B ) } )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   {cab 2608   A.wral 2912   [.wsbc 3435   [_csb 3533   {csn 4177   Fn wfn 5883   ` cfv 5888   X_cixp 7908

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-nul 4789  ax-pow 4843

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-dm 5124  df-iota 5851  df-fn 5891  df-fv 5896  df-ixp 7909

This theorem is referenced by:  ixpsnbasval  19209