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Theorem uniixp 6839
Description: The union of an infinite Cartesian product is included in a cross product. (Contributed by NM, 28-Sep-2006.) (Revised by Mario Carneiro, 24-Jun-2015.)
Assertion
Ref Expression
uniixp  |-  U. X_ x  e.  A  B  C_  ( A  X.  U_ x  e.  A  B )
Distinct variable group:    x, A
Allowed substitution hint:    B( x)

Proof of Theorem uniixp
Dummy variable  f is distinct from all other variables.
StepHypRef Expression
1 ixpf 6838 . . . . 5  |-  ( f  e.  X_ x  e.  A  B  ->  f : A --> U_ x  e.  A  B
)
2 fssxp 5400 . . . . 5  |-  ( f : A --> U_ x  e.  A  B  ->  f 
C_  ( A  X.  U_ x  e.  A  B
) )
31, 2syl 15 . . . 4  |-  ( f  e.  X_ x  e.  A  B  ->  f  C_  ( A  X.  U_ x  e.  A  B ) )
4 vex 2791 . . . . 5  |-  f  e. 
_V
54elpw 3631 . . . 4  |-  ( f  e.  ~P ( A  X.  U_ x  e.  A  B )  <->  f  C_  ( A  X.  U_ x  e.  A  B )
)
63, 5sylibr 203 . . 3  |-  ( f  e.  X_ x  e.  A  B  ->  f  e.  ~P ( A  X.  U_ x  e.  A  B )
)
76ssriv 3184 . 2  |-  X_ x  e.  A  B  C_  ~P ( A  X.  U_ x  e.  A  B )
8 sspwuni 3987 . 2  |-  ( X_ x  e.  A  B  C_ 
~P ( A  X.  U_ x  e.  A  B
)  <->  U. X_ x  e.  A  B  C_  ( A  X.  U_ x  e.  A  B
) )
97, 8mpbi 199 1  |-  U. X_ x  e.  A  B  C_  ( A  X.  U_ x  e.  A  B )
Colors of variables: wff set class
Syntax hints:    e. wcel 1684    C_ wss 3152   ~Pcpw 3625   U.cuni 3827   U_ciun 3905    X. cxp 4687   -->wf 5251   X_cixp 6817
This theorem is referenced by:  ixpexg  6840
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fv 5263  df-ixp 6818
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