Theorem elixp2 6761

Description: Membership in an infinite Cartesian product. See df-ixp 6758 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 5346	. . . . 5 |- ( f = F -> ( f Fn A <-> F Fn A ) )
2		fveq1 5557	. . . . . . 7 |- ( f = F -> ( f ` x ) = ( F ` x ) )
3	2	eleq1d 2265	. . . . . 6 |- ( f = F -> ( ( f ` x ) e. B <-> ( F ` x ) e. B ) )
4	3	ralbidv 2497	. . . . 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 6759	. . . 4 |- X_ x e. A B = { f | ( f Fn A /\ A. x e. A ( f ` x ) e. B ) }
7	5, 6	elab2g 2911	. . 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 2774	. . 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 984	. 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 980   = wceq 1364   e. wcel 2167   A.wral 2475   _Vcvv 2763   Fn wfn 5253   ` cfv 5258   X_cixp 6757

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-v 2765  df-un 3161  df-in 3163  df-ss 3170  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-br 4034  df-opab 4095  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-iota 5219  df-fun 5260  df-fn 5261  df-fv 5266  df-ixp 6758

This theorem is referenced by:  fvixp  6762  ixpfn  6763  elixp  6764  ixpf  6779  resixp  6792  mptelixpg  6793  xpsfrnel  12987  xpscf  12990