Theorem ixpf 6677

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 6659	. 2 |- ( F e. X_ x e. A B <-> ( F e. _V /\ F Fn A /\ A. x e. A ( F ` x ) e. B ) )
2		ssiun2 3903	. . . . . . 7 |- ( x e. A -> B C_ U_ x e. A B )
3	2	sseld 3136	. . . . . 6 |- ( x e. A -> ( ( F ` x ) e. B -> ( F ` x ) e. U_ x e. A B ) )
4	3	ralimia 2525	. . . . 5 |- ( A. x e. A ( F ` x ) e. B -> A. x e. A ( F ` x ) e. U_ x e. A B )
5	4	anim2i 340	. . . 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 2306	. . . . 5 |- F/_ x A
7		nfiu1 3890	. . . . 5 |- F/_ x U_ x e. A B
8		nfcv 2306	. . . . 5 |- F/_ x F
9	6, 7, 8	ffnfvf 5638	. . . 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 1004	. 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 967   e. wcel 2135   A.wral 2442   _Vcvv 2721   U_ciun 3860   Fn wfn 5177   -->wf 5178   ` cfv 5182   X_cixp 6655

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 699  ax-5 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-14 2138  ax-ext 2146  ax-sep 4094  ax-pow 4147  ax-pr 4181

This theorem depends on definitions:  df-bi 116  df-3an 969  df-tru 1345  df-nf 1448  df-sb 1750  df-eu 2016  df-mo 2017  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-ral 2447  df-rex 2448  df-v 2723  df-sbc 2947  df-un 3115  df-in 3117  df-ss 3124  df-pw 3555  df-sn 3576  df-pr 3577  df-op 3579  df-uni 3784  df-iun 3862  df-br 3977  df-opab 4038  df-mpt 4039  df-id 4265  df-xp 4604  df-rel 4605  df-cnv 4606  df-co 4607  df-dm 4608  df-rn 4609  df-iota 5147  df-fun 5184  df-fn 5185  df-f 5186  df-fv 5190  df-ixp 6656

This theorem is referenced by:  uniixp  6678  ixpssmap2g  6684