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Theorem ixpf 7087
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
StepHypRef Expression
1 elixp2 7069 . 2  |-  ( F  e.  X_ x  e.  A  B 
<->  ( F  e.  _V  /\  F  Fn  A  /\  A. x  e.  A  ( F `  x )  e.  B ) )
2 ssiun2 4136 . . . . . . 7  |-  ( x  e.  A  ->  B  C_ 
U_ x  e.  A  B )
32sseld 3349 . . . . . 6  |-  ( x  e.  A  ->  (
( F `  x
)  e.  B  -> 
( F `  x
)  e.  U_ x  e.  A  B )
)
43ralimia 2781 . . . . 5  |-  ( A. x  e.  A  ( F `  x )  e.  B  ->  A. x  e.  A  ( F `  x )  e.  U_ x  e.  A  B
)
54anim2i 554 . . . 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 2574 . . . . 5  |-  F/_ x A
7 nfiu1 4123 . . . . 5  |-  F/_ x U_ x  e.  A  B
8 nfcv 2574 . . . . 5  |-  F/_ x F
96, 7, 8ffnfvf 5898 . . . 4  |-  ( F : A --> U_ x  e.  A  B  <->  ( F  Fn  A  /\  A. x  e.  A  ( F `  x )  e.  U_ x  e.  A  B
) )
105, 9sylibr 205 . . 3  |-  ( ( F  Fn  A  /\  A. x  e.  A  ( F `  x )  e.  B )  ->  F : A --> U_ x  e.  A  B )
11103adant1 976 . 2  |-  ( ( F  e.  _V  /\  F  Fn  A  /\  A. x  e.  A  ( F `  x )  e.  B )  ->  F : A --> U_ x  e.  A  B )
121, 11sylbi 189 1  |-  ( F  e.  X_ x  e.  A  B  ->  F : A --> U_ x  e.  A  B
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    /\ w3a 937    e. wcel 1726   A.wral 2707   _Vcvv 2958   U_ciun 4095    Fn wfn 5452   -->wf 5453   ` cfv 5457   X_cixp 7066
This theorem is referenced by:  uniixp  7088  ixpssmap2g  7094
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pr 4406
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-fv 5465  df-ixp 7067
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