Theorem ixpf 6867

Description: A member of an infinite Cartesian product maps to the indexed union of the product argument. Remark in [Enderton] p. 54. (Contributed by NM, 28-Sep-2006.)

Assertion

Ref	Expression
ixpf	|- ( F e. X_ x e. A B -> F : A --> U_ x e. A B )

Distinct variable groups:   x, A   x, F

Allowed substitution hint:   B( x)

Proof of Theorem ixpf

Step	Hyp	Ref	Expression
1		elixp2 6849	. 2 |- ( F e. X_ x e. A B <-> ( F e. _V /\ F Fn A /\ A. x e. A ( F ` x ) e. B ) )
2		ssiun2 4008	. . . . . . 7 |- ( x e. A -> B C_ U_ x e. A B )
3	2	sseld 3223	. . . . . 6 |- ( x e. A -> ( ( F ` x ) e. B -> ( F ` x ) e. U_ x e. A B ) )
4	3	ralimia 2591	. . . . 5 |- ( A. x e. A ( F ` x ) e. B -> A. x e. A ( F ` x ) e. U_ x e. A B )
5	4	anim2i 342	. . . 4 |- ( ( F Fn A /\ A. x e. A ( F ` x ) e. B ) -> ( F Fn A /\ A. x e. A ( F ` x ) e. U_ x e. A B ) )
6		nfcv 2372	. . . . 5 |- F/_ x A
7		nfiu1 3995	. . . . 5 |- F/_ x U_ x e. A B
8		nfcv 2372	. . . . 5 |- F/_ x F
9	6, 7, 8	ffnfvf 5794	. . . 4 |- ( F : A --> U_ x e. A B <-> ( F Fn A /\ A. x e. A ( F ` x ) e. U_ x e. A B ) )
10	5, 9	sylibr 134	. . 3 |- ( ( F Fn A /\ A. x e. A ( F ` x ) e. B ) -> F : A --> U_ x e. A B )
11	10	3adant1 1039	. 2 |- ( ( F e. _V /\ F Fn A /\ A. x e. A ( F ` x ) e. B ) -> F : A --> U_ x e. A B )
12	1, 11	sylbi 121	1 |- ( F e. X_ x e. A B -> F : A --> U_ x e. A B )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 1002   e. wcel 2200   A.wral 2508   _Vcvv 2799   U_ciun 3965   Fn wfn 5313   -->wf 5314   ` cfv 5318   X_cixp 6845

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-sbc 3029  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-id 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-fv 5326  df-ixp 6846

This theorem is referenced by:  uniixp  6868  ixpssmap2g  6874