Theorem elixp2 6939

Description: Membership in an infinite Cartesian product. See df-ixp 6936 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 5446	. . . . 5 |- ( f = F -> ( f Fn A <-> F Fn A ) )
2		fveq1 5671	. . . . . . 7 |- ( f = F -> ( f ` x ) = ( F ` x ) )
3	2	eleq1d 2303	. . . . . 6 |- ( f = F -> ( ( f ` x ) e. B <-> ( F ` x ) e. B ) )
4	3	ralbidv 2544	. . . . 5 |- ( f = F -> ( A. x e. A ( f ` x ) e. B <-> A. x e. A ( F ` x ) e. B ) )
5	1, 4	anbi12d 473	. . . 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 6937	. . . 4 |- X_ x e. A B = { f | ( f Fn A /\ A. x e. A ( f ` x ) e. B ) }
7	5, 6	elab2g 2966	. . 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 454	. 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 2827	. . 3 |- ( F e. X_ x e. A B -> F e. _V )
10	9	pm4.71ri 392	. 2 |- ( F e. X_ x e. A B <-> ( F e. _V /\ F e. X_ x e. A B ) )
11		3anass 1009	. 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 ) )

Syntax hints:   /\ wa 104   <-> wb 105   /\ w3a 1005   = wceq 1398   e. wcel 2205   A.wral 2522   _Vcvv 2815   Fn wfn 5349   ` cfv 5354   X_cixp 6935

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2216

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-v 2817  df-un 3217  df-in 3219  df-ss 3226  df-sn 3697  df-pr 3698  df-op 3700  df-uni 3917  df-br 4112  df-opab 4174  df-rel 4758  df-cnv 4759  df-co 4760  df-dm 4761  df-iota 5314  df-fun 5356  df-fn 5357  df-fv 5362  df-ixp 6936

This theorem is referenced by:  fvixp  6940  ixpfn  6941  elixp  6942  ixpf  6957  resixp  6970  mptelixpg  6971  prdsbasprj  13512  xpsfrnel  13574  xpscf  13577  depindlem2  16519