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Theorem marypha2lem1 7440
Description: Lemma for marypha2 7444. 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 4132 . . 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 3942 . . . 4  |-  ( x  e.  A  ->  { x }  C_  A )
4 fvssunirn 5754 . . . 4  |-  ( F `
 x )  C_  U.
ran  F
5 xpss12 4981 . . . 4  |-  ( ( { x }  C_  A  /\  ( F `  x )  C_  U. ran  F )  ->  ( {
x }  X.  ( F `  x )
)  C_  ( A  X.  U. ran  F ) )
63, 4, 5sylancl 644 . . 3  |-  ( x  e.  A  ->  ( { x }  X.  ( F `  x ) )  C_  ( A  X.  U. ran  F ) )
72, 6mprgbir 2776 . 2  |-  U_ x  e.  A  ( {
x }  X.  ( F `  x )
)  C_  ( A  X.  U. ran  F )
81, 7eqsstri 3378 1  |-  T  C_  ( A  X.  U. ran  F )
Colors of variables: wff set class
Syntax hints:    = wceq 1652    e. wcel 1725    C_ wss 3320   {csn 3814   U.cuni 4015   U_ciun 4093    X. cxp 4876   ran crn 4879   ` cfv 5454
This theorem is referenced by:  marypha2  7444
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-xp 4884  df-cnv 4886  df-dm 4888  df-rn 4889  df-iota 5418  df-fv 5462
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