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Theorem ixpf 6713
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 6695 . 2  |-  ( F  e.  X_ x  e.  A  B 
<->  ( F  e.  _V  /\  F  Fn  A  /\  A. x  e.  A  ( F `  x )  e.  B ) )
2 ssiun2 3927 . . . . . . 7  |-  ( x  e.  A  ->  B  C_ 
U_ x  e.  A  B )
32sseld 3154 . . . . . 6  |-  ( x  e.  A  ->  (
( F `  x
)  e.  B  -> 
( F `  x
)  e.  U_ x  e.  A  B )
)
43ralimia 2538 . . . . 5  |-  ( A. x  e.  A  ( F `  x )  e.  B  ->  A. x  e.  A  ( F `  x )  e.  U_ x  e.  A  B
)
54anim2i 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 2319 . . . . 5  |-  F/_ x A
7 nfiu1 3914 . . . . 5  |-  F/_ x U_ x  e.  A  B
8 nfcv 2319 . . . . 5  |-  F/_ x F
96, 7, 8ffnfvf 5670 . . . 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 134 . . 3  |-  ( ( F  Fn  A  /\  A. x  e.  A  ( F `  x )  e.  B )  ->  F : A --> U_ x  e.  A  B )
11103adant1 1015 . 2  |-  ( ( F  e.  _V  /\  F  Fn  A  /\  A. x  e.  A  ( F `  x )  e.  B )  ->  F : A --> U_ x  e.  A  B )
121, 11sylbi 121 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 104    /\ w3a 978    e. wcel 2148   A.wral 2455   _Vcvv 2737   U_ciun 3884    Fn wfn 5206   -->wf 5207   ` cfv 5211   X_cixp 6691
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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171  ax-pr 4205
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-sbc 2963  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-id 4289  df-xp 4628  df-rel 4629  df-cnv 4630  df-co 4631  df-dm 4632  df-rn 4633  df-iota 5173  df-fun 5213  df-fn 5214  df-f 5215  df-fv 5219  df-ixp 6692
This theorem is referenced by:  uniixp  6714  ixpssmap2g  6720
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