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Theorem marypha2lem1 7204
Description: Lemma for marypha2 7208. Properties of the used relation. (Contributed by Stefan O'Rear, 20-Feb-2015.)
Hypothesis
Ref Expression
marypha2lem.t  |-  T  = 
U_ x  e.  A  ( { x }  X.  ( F `  x ) )
Assertion
Ref Expression
marypha2lem1  |-  T  C_  ( A  X.  U. ran  F )
Distinct variable groups:    x, A    x, F
Allowed substitution hint:    T( x)

Proof of Theorem marypha2lem1
StepHypRef Expression
1 marypha2lem.t . 2  |-  T  = 
U_ x  e.  A  ( { x }  X.  ( F `  x ) )
2 iunss 3959 . . 3  |-  ( U_ x  e.  A  ( { x }  X.  ( F `  x ) )  C_  ( A  X.  U. ran  F )  <->  A. x  e.  A  ( { x }  X.  ( F `  x ) )  C_  ( A  X.  U. ran  F ) )
3 snssi 3775 . . . 4  |-  ( x  e.  A  ->  { x }  C_  A )
4 fvssunirn 5567 . . . 4  |-  ( F `
 x )  C_  U.
ran  F
5 xpss12 4808 . . . 4  |-  ( ( { x }  C_  A  /\  ( F `  x )  C_  U. ran  F )  ->  ( {
x }  X.  ( F `  x )
)  C_  ( A  X.  U. ran  F ) )
63, 4, 5sylancl 643 . . 3  |-  ( x  e.  A  ->  ( { x }  X.  ( F `  x ) )  C_  ( A  X.  U. ran  F ) )
72, 6mprgbir 2626 . 2  |-  U_ x  e.  A  ( {
x }  X.  ( F `  x )
)  C_  ( A  X.  U. ran  F )
81, 7eqsstri 3221 1  |-  T  C_  ( A  X.  U. ran  F )
Colors of variables: wff set class
Syntax hints:    = wceq 1632    e. wcel 1696    C_ wss 3165   {csn 3653   U.cuni 3843   U_ciun 3921    X. cxp 4703   ran crn 4706   ` cfv 5271
This theorem is referenced by:  marypha2  7208
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-13 1698  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-pow 4204  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-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-xp 4711  df-cnv 4713  df-dm 4715  df-rn 4716  df-iota 5235  df-fv 5279
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