Theorem elixp2 6680

Description: Membership in an infinite Cartesian product. See df-ixp 6677 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 5286	. . . . 5 |- ( f = F -> ( f Fn A <-> F Fn A ) )
2		fveq1 5495	. . . . . . 7 |- ( f = F -> ( f ` x ) = ( F ` x ) )
3	2	eleq1d 2239	. . . . . 6 |- ( f = F -> ( ( f ` x ) e. B <-> ( F ` x ) e. B ) )
4	3	ralbidv 2470	. . . . 5 |- ( f = F -> ( A. x e. A ( f ` x ) e. B <-> A. x e. A ( F ` x ) e. B ) )
5	1, 4	anbi12d 470	. . . 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 6678	. . . 4 |- X_ x e. A B = { f | ( f Fn A /\ A. x e. A ( f ` x ) e. B ) }
7	5, 6	elab2g 2877	. . 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 451	. 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 2741	. . 3 |- ( F e. X_ x e. A B -> F e. _V )
10	9	pm4.71ri 390	. 2 |- ( F e. X_ x e. A B <-> ( F e. _V /\ F e. X_ x e. A B ) )
11		3anass 977	. 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 211	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 103   <-> wb 104   /\ w3a 973   = wceq 1348   e. wcel 2141   A.wral 2448   _Vcvv 2730   Fn wfn 5193   ` cfv 5198   X_cixp 6676

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-iota 5160  df-fun 5200  df-fn 5201  df-fv 5206  df-ixp 6677

This theorem is referenced by:  fvixp  6681  ixpfn  6682  elixp  6683  ixpf  6698  resixp  6711  mptelixpg  6712