Theorem elixp 7915

Description: Membership in an infinite Cartesian product. (Contributed by NM, 28-Sep-2006.)

Hypothesis

Ref	Expression
elixp.1	|- F e. _V

Assertion

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

Distinct variable groups:   x, F   x, A

Allowed substitution hint:   B( x)

Proof of Theorem elixp

Step	Hyp	Ref	Expression
1		elixp2 7912	. 2 |- ( F e. X_ x e. A B <-> ( F e. _V /\ F Fn A /\ A. x e. A ( F ` x ) e. B ) )
2		elixp.1	. . 3 |- F e. _V
3		3anass 1042	. . 3 |- ( ( 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 ) ) )
4	2, 3	mpbiran 953	. 2 |- ( ( F e. _V /\ F Fn A /\ A. x e. A ( F ` x ) e. B ) <-> ( F Fn A /\ A. x e. A ( F ` x ) e. B ) )
5	1, 4	bitri 264	1 |- ( F e. X_ x e. A B <-> ( F Fn A /\ A. x e. A ( F ` x ) e. B ) )

Syntax hints:   <-> wb 196   /\ wa 384   /\ w3a 1037   e. wcel 1990   A.wral 2912   _Vcvv 3200   Fn wfn 5883   ` cfv 5888   X_cixp 7908

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fn 5891  df-fv 5896  df-ixp 7909

This theorem is referenced by:  elixpconst  7916  ixpin  7933  ixpiin  7934  resixpfo  7946  elixpsn  7947  boxriin  7950  boxcutc  7951  ixpfi2  8264  ixpiunwdom  8496  dfac9  8958  ac9  9305  ac9s  9315  konigthlem  9390  xpscf  16226  cofucl  16548  yonedalem3  16920  psrbaglefi  19372  ptpjpre1  21374  ptpjcn  21414  ptpjopn  21415  ptclsg  21418  dfac14  21421  pthaus  21441  xkopt  21458  ptcmplem2  21857  ptcmplem3  21858  ptcmplem4  21859  prdsbl  22296  prdsxmslem2  22334  eulerpartlemb  30430  ptpconn  31215  finixpnum  33394  ptrest  33408  poimirlem29  33438  poimirlem30  33439  inixp  33523  prdstotbnd  33593  ioorrnopnlem  40524  hoicvr  40762  hoidmvlelem3  40811  hspdifhsp  40830  hspmbllem2  40841