Theorem ixpf 6698

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 6680	. 2 |- ( F e. X_ x e. A B <-> ( F e. _V /\ F Fn A /\ A. x e. A ( F ` x ) e. B ) )
2		ssiun2 3916	. . . . . . 7 |- ( x e. A -> B C_ U_ x e. A B )
3	2	sseld 3146	. . . . . 6 |- ( x e. A -> ( ( F ` x ) e. B -> ( F ` x ) e. U_ x e. A B ) )
4	3	ralimia 2531	. . . . 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 2312	. . . . 5 |- F/_ x A
7		nfiu1 3903	. . . . 5 |- F/_ x U_ x e. A B
8		nfcv 2312	. . . . 5 |- F/_ x F
9	6, 7, 8	ffnfvf 5655	. . . 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 1010	. 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 973   e. wcel 2141   A.wral 2448   _Vcvv 2730   U_ciun 3873   Fn wfn 5193   -->wf 5194   ` cfv 5198   X_cixp 6676

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-sbc 2956  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-fv 5206  df-ixp 6677

This theorem is referenced by:  uniixp  6699  ixpssmap2g  6705