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Theorem ixpf 7043
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 7025 . 2  |-  ( F  e.  X_ x  e.  A  B 
<->  ( F  e.  _V  /\  F  Fn  A  /\  A. x  e.  A  ( F `  x )  e.  B ) )
2 ssiun2 4094 . . . . . . 7  |-  ( x  e.  A  ->  B  C_ 
U_ x  e.  A  B )
32sseld 3307 . . . . . 6  |-  ( x  e.  A  ->  (
( F `  x
)  e.  B  -> 
( F `  x
)  e.  U_ x  e.  A  B )
)
43ralimia 2739 . . . . 5  |-  ( A. x  e.  A  ( F `  x )  e.  B  ->  A. x  e.  A  ( F `  x )  e.  U_ x  e.  A  B
)
54anim2i 553 . . . 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 2540 . . . . 5  |-  F/_ x A
7 nfiu1 4081 . . . . 5  |-  F/_ x U_ x  e.  A  B
8 nfcv 2540 . . . . 5  |-  F/_ x F
96, 7, 8ffnfvf 5854 . . . 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 204 . . 3  |-  ( ( F  Fn  A  /\  A. x  e.  A  ( F `  x )  e.  B )  ->  F : A --> U_ x  e.  A  B )
11103adant1 975 . 2  |-  ( ( F  e.  _V  /\  F  Fn  A  /\  A. x  e.  A  ( F `  x )  e.  B )  ->  F : A --> U_ x  e.  A  B )
121, 11sylbi 188 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 359    /\ w3a 936    e. wcel 1721   A.wral 2666   _Vcvv 2916   U_ciun 4053    Fn wfn 5408   -->wf 5409   ` cfv 5413   X_cixp 7022
This theorem is referenced by:  uniixp  7044  ixpssmap2g  7050
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pr 4363
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-sbc 3122  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-fv 5421  df-ixp 7023
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