Theorem elixp2 6984

Description: Membership in an infinite Cartesian product. See df-ixp 6981 for discussion of the notation. (Contributed by NM, 28-Sep-2006.)

Assertion

Ref	Expression
elixp2	|- ( F e. X_ x e. A B <-> ( F e. _V /\ F Fn A /\ A. x e. A ( F ` x ) e. B ) )

Distinct variable groups:   x, A   x, F

Allowed substitution hint:   B( x)

Proof of Theorem elixp2

Dummy variable f is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fneq1 5469	. . . . 5 |- ( f = F -> ( f Fn A <-> F Fn A ) )
2		fveq1 5694	. . . . . . 7 |- ( f = F -> ( f ` x ) = ( F ` x ) )
3	2	eleq1d 2307	. . . . . 6 |- ( f = F -> ( ( f ` x ) e. B <-> ( F ` x ) e. B ) )
4	3	ralbidv 2550	. . . . 5 |- ( f = F -> ( A. x e. A ( f ` x ) e. B <-> A. x e. A ( F ` x ) e. B ) )
5	1, 4	anbi12d 477	. . . 4 |- ( f = F -> ( ( f Fn A /\ A. x e. A ( f ` x ) e. B ) <-> ( F Fn A /\ A. x e. A ( F ` x ) e. B ) ) )
6		dfixp 6982	. . . 4 |- X_ x e. A B = { f | ( f Fn A /\ A. x e. A ( f ` x ) e. B ) }
7	5, 6	elab2g 2973	. . 3 |- ( F e. _V -> ( F e. X_ x e. A B <-> ( F Fn A /\ A. x e. A ( F ` x ) e. B ) ) )
8	7	pm5.32i 458	. 2 |- ( ( F e. _V /\ F e. X_ x e. A B ) <-> ( F e. _V /\ ( F Fn A /\ A. x e. A ( F ` x ) e. B ) ) )
9		elex 2833	. . 3 |- ( F e. X_ x e. A B -> F e. _V )
10	9	pm4.71ri 396	. 2 |- ( F e. X_ x e. A B <-> ( F e. _V /\ F e. X_ x e. A B ) )
11		3anass 1013	. 2 |- ( ( F e. _V /\ F Fn A /\ A. x e. A ( F ` x ) e. B ) <-> ( F e. _V /\ ( F Fn A /\ A. x e. A ( F ` x ) e. B ) ) )
12	8, 10, 11	3bitr4i 212	1 |- ( F e. X_ x e. A B <-> ( F e. _V /\ F Fn A /\ A. x e. A ( F ` x ) e. B ) )

This proof depends on syntax axioms:   /\ wa 104   <-> wb 105   /\ w3a 1009   = wceq 1402   e. wcel 2209   A.wral 2528   _Vcvv 2821   Fn wfn 5372   ` cfv 5377   X_cixp 6980

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-br 4131  df-opab 4193  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-iota 5337  df-fun 5379  df-fn 5380  df-fv 5385  df-ixp 6981

This theorem is used by:  fvixp  6985  ixpfn  6986  elixp  6987  ixpf  7002  resixp  7015  mptelixpg  7016  xpsfrnel  13665  xpscf  13668  prdsbasprj  14182  depindlem2  16748