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Theorem marypha2lem1 7188
Description: Lemma for marypha2 7192. 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 3943 . . 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 3759 . . . 4  |-  ( x  e.  A  ->  { x }  C_  A )
4 fvssunirn 5551 . . . 4  |-  ( F `
 x )  C_  U.
ran  F
5 xpss12 4792 . . . 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 2613 . 2  |-  U_ x  e.  A  ( {
x }  X.  ( F `  x )
)  C_  ( A  X.  U. ran  F )
81, 7eqsstri 3208 1  |-  T  C_  ( A  X.  U. ran  F )
Colors of variables: wff set class
Syntax hints:    = wceq 1623    e. wcel 1684    C_ wss 3152   {csn 3640   U.cuni 3827   U_ciun 3905    X. cxp 4687   ran crn 4690   ` cfv 5255
This theorem is referenced by:  marypha2  7192
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-13 1686  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-pow 4188  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-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-xp 4695  df-cnv 4697  df-dm 4699  df-rn 4700  df-iota 5219  df-fv 5263
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