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Theorem uniixp 6855
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 6854 . . . . 5  |-  ( f  e.  X_ x  e.  A  B  ->  f : A --> U_ x  e.  A  B
)
2 fssxp 5416 . . . . 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 2804 . . . . 5  |-  f  e. 
_V
54elpw 3644 . . . 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 3197 . 2  |-  X_ x  e.  A  B  C_  ~P ( A  X.  U_ x  e.  A  B )
8 sspwuni 4003 . 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 1696    C_ wss 3165   ~Pcpw 3638   U.cuni 3843   U_ciun 3921    X. cxp 4703   -->wf 5267   X_cixp 6833
This theorem is referenced by:  ixpexg  6856
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-pw 3640  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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