Theorem ixpf 6607

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 6589	. 2 |- ( F e. X_ x e. A B <-> ( F e. _V /\ F Fn A /\ A. x e. A ( F ` x ) e. B ) )
2		ssiun2 3851	. . . . . . 7 |- ( x e. A -> B C_ U_ x e. A B )
3	2	sseld 3091	. . . . . 6 |- ( x e. A -> ( ( F ` x ) e. B -> ( F ` x ) e. U_ x e. A B ) )
4	3	ralimia 2491	. . . . 5 |- ( A. x e. A ( F ` x ) e. B -> A. x e. A ( F ` x ) e. U_ x e. A B )
5	4	anim2i 339	. . . 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 2279	. . . . 5 |- F/_ x A
7		nfiu1 3838	. . . . 5 |- F/_ x U_ x e. A B
8		nfcv 2279	. . . . 5 |- F/_ x F
9	6, 7, 8	ffnfvf 5572	. . . 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 133	. . 3 |- ( ( F Fn A /\ A. x e. A ( F ` x ) e. B ) -> F : A --> U_ x e. A B )
11	10	3adant1 999	. 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 120	1 |- ( F e. X_ x e. A B -> F : A --> U_ x e. A B )

Syntax hints:   -> wi 4   /\ wa 103   /\ w3a 962   e. wcel 1480   A.wral 2414   _Vcvv 2681   U_ciun 3808   Fn wfn 5113   -->wf 5114   ` cfv 5118   X_cixp 6585

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-v 2683  df-sbc 2905  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-iun 3810  df-br 3925  df-opab 3985  df-mpt 3986  df-id 4210  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-iota 5083  df-fun 5120  df-fn 5121  df-f 5122  df-fv 5126  df-ixp 6586

This theorem is referenced by:  uniixp  6608  ixpssmap2g  6614