Theorem ixpf 6779

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 6761	. 2 |- ( F e. X_ x e. A B <-> ( F e. _V /\ F Fn A /\ A. x e. A ( F ` x ) e. B ) )
2		ssiun2 3959	. . . . . . 7 |- ( x e. A -> B C_ U_ x e. A B )
3	2	sseld 3182	. . . . . 6 |- ( x e. A -> ( ( F ` x ) e. B -> ( F ` x ) e. U_ x e. A B ) )
4	3	ralimia 2558	. . . . 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 2339	. . . . 5 |- F/_ x A
7		nfiu1 3946	. . . . 5 |- F/_ x U_ x e. A B
8		nfcv 2339	. . . . 5 |- F/_ x F
9	6, 7, 8	ffnfvf 5721	. . . 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 1017	. 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 980   e. wcel 2167   A.wral 2475   _Vcvv 2763   U_ciun 3916   Fn wfn 5253   -->wf 5254   ` cfv 5258   X_cixp 6757

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207  ax-pr 4242

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-v 2765  df-sbc 2990  df-un 3161  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-iun 3918  df-br 4034  df-opab 4095  df-mpt 4096  df-id 4328  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-fv 5266  df-ixp 6758

This theorem is referenced by:  uniixp  6780  ixpssmap2g  6786