Theorem elixp2 6756

Description: Membership in an infinite Cartesian product. See df-ixp 6753 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 5342	. . . . 5 |- ( f = F -> ( f Fn A <-> F Fn A ) )
2		fveq1 5553	. . . . . . 7 |- ( f = F -> ( f ` x ) = ( F ` x ) )
3	2	eleq1d 2262	. . . . . 6 |- ( f = F -> ( ( f ` x ) e. B <-> ( F ` x ) e. B ) )
4	3	ralbidv 2494	. . . . 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 6754	. . . 4 |- X_ x e. A B = { f | ( f Fn A /\ A. x e. A ( f ` x ) e. B ) }
7	5, 6	elab2g 2907	. . 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 2771	. . 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 2164   A.wral 2472   _Vcvv 2760   Fn wfn 5249   ` cfv 5254   X_cixp 6752

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-un 3157  df-in 3159  df-ss 3166  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-br 4030  df-opab 4091  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-iota 5215  df-fun 5256  df-fn 5257  df-fv 5262  df-ixp 6753

This theorem is referenced by:  fvixp  6757  ixpfn  6758  elixp  6759  ixpf  6774  resixp  6787  mptelixpg  6788  xpsfrnel  12927  xpscf  12930