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Theorem ixpf 6854
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 6836 . 2  |-  ( F  e.  X_ x  e.  A  B 
<->  ( F  e.  _V  /\  F  Fn  A  /\  A. x  e.  A  ( F `  x )  e.  B ) )
2 ssiun2 3961 . . . . . . 7  |-  ( x  e.  A  ->  B  C_ 
U_ x  e.  A  B )
32sseld 3192 . . . . . 6  |-  ( x  e.  A  ->  (
( F `  x
)  e.  B  -> 
( F `  x
)  e.  U_ x  e.  A  B )
)
43ralimia 2629 . . . . 5  |-  ( A. x  e.  A  ( F `  x )  e.  B  ->  A. x  e.  A  ( F `  x )  e.  U_ x  e.  A  B
)
54anim2i 552 . . . 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 2432 . . . . 5  |-  F/_ x A
7 nfiu1 3949 . . . . 5  |-  F/_ x U_ x  e.  A  B
8 nfcv 2432 . . . . 5  |-  F/_ x F
96, 7, 8ffnfvf 5702 . . . 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 203 . . 3  |-  ( ( F  Fn  A  /\  A. x  e.  A  ( F `  x )  e.  B )  ->  F : A --> U_ x  e.  A  B )
11103adant1 973 . 2  |-  ( ( F  e.  _V  /\  F  Fn  A  /\  A. x  e.  A  ( F `  x )  e.  B )  ->  F : A --> U_ x  e.  A  B )
121, 11sylbi 187 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 358    /\ w3a 934    e. wcel 1696   A.wral 2556   _Vcvv 2801   U_ciun 3921    Fn wfn 5266   -->wf 5267   ` cfv 5271   X_cixp 6833
This theorem is referenced by:  uniixp  6855  ixpssmap2g  6861  npincppr  25262
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-fv 5279  df-ixp 6834
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