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Theorem ixpf 7490
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 7472 . 2  |-  ( F  e.  X_ x  e.  A  B 
<->  ( F  e.  _V  /\  F  Fn  A  /\  A. x  e.  A  ( F `  x )  e.  B ) )
2 ssiun2 4355 . . . . . . 7  |-  ( x  e.  A  ->  B  C_ 
U_ x  e.  A  B )
32sseld 3486 . . . . . 6  |-  ( x  e.  A  ->  (
( F `  x
)  e.  B  -> 
( F `  x
)  e.  U_ x  e.  A  B )
)
43ralimia 2832 . . . . 5  |-  ( A. x  e.  A  ( F `  x )  e.  B  ->  A. x  e.  A  ( F `  x )  e.  U_ x  e.  A  B
)
54anim2i 569 . . . 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 2603 . . . . 5  |-  F/_ x A
7 nfiu1 4342 . . . . 5  |-  F/_ x U_ x  e.  A  B
8 nfcv 2603 . . . . 5  |-  F/_ x F
96, 7, 8ffnfvf 6040 . . . 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 212 . . 3  |-  ( ( F  Fn  A  /\  A. x  e.  A  ( F `  x )  e.  B )  ->  F : A --> U_ x  e.  A  B )
11103adant1 1013 . 2  |-  ( ( F  e.  _V  /\  F  Fn  A  /\  A. x  e.  A  ( F `  x )  e.  B )  ->  F : A --> U_ x  e.  A  B )
121, 11sylbi 195 1  |-  ( F  e.  X_ x  e.  A  B  ->  F : A --> U_ x  e.  A  B
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 972    e. wcel 1802   A.wral 2791   _Vcvv 3093   U_ciun 4312    Fn wfn 5570   -->wf 5571   ` cfv 5575   X_cixp 7468
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-sep 4555  ax-nul 4563  ax-pr 4673
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-ral 2796  df-rex 2797  df-rab 2800  df-v 3095  df-sbc 3312  df-dif 3462  df-un 3464  df-in 3466  df-ss 3473  df-nul 3769  df-if 3924  df-sn 4012  df-pr 4014  df-op 4018  df-uni 4232  df-iun 4314  df-br 4435  df-opab 4493  df-mpt 4494  df-id 4782  df-xp 4992  df-rel 4993  df-cnv 4994  df-co 4995  df-dm 4996  df-rn 4997  df-iota 5538  df-fun 5577  df-fn 5578  df-f 5579  df-fv 5583  df-ixp 7469
This theorem is referenced by:  uniixp  7491  ixpssmap2g  7497
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