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Theorem marypha2lem2 7691
Description: Lemma for marypha2 7694. 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
marypha2lem2  |-  T  =  { <. x ,  y
>.  |  ( x  e.  A  /\  y  e.  ( F `  x
) ) }
Distinct variable groups:    x, A, y    x, F, y
Allowed substitution hints:    T( x, y)

Proof of Theorem marypha2lem2
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 marypha2lem.t . 2  |-  T  = 
U_ x  e.  A  ( { x }  X.  ( F `  x ) )
2 sneq 3892 . . . 4  |-  ( x  =  z  ->  { x }  =  { z } )
3 fveq2 5696 . . . 4  |-  ( x  =  z  ->  ( F `  x )  =  ( F `  z ) )
42, 3xpeq12d 4870 . . 3  |-  ( x  =  z  ->  ( { x }  X.  ( F `  x ) )  =  ( { z }  X.  ( F `  z )
) )
54cbviunv 4214 . 2  |-  U_ x  e.  A  ( {
x }  X.  ( F `  x )
)  =  U_ z  e.  A  ( {
z }  X.  ( F `  z )
)
6 df-xp 4851 . . . . 5  |-  ( { z }  X.  ( F `  z )
)  =  { <. x ,  y >.  |  ( x  e.  { z }  /\  y  e.  ( F `  z
) ) }
76a1i 11 . . . 4  |-  ( z  e.  A  ->  ( { z }  X.  ( F `  z ) )  =  { <. x ,  y >.  |  ( x  e.  { z }  /\  y  e.  ( F `  z
) ) } )
87iuneq2i 4194 . . 3  |-  U_ z  e.  A  ( {
z }  X.  ( F `  z )
)  =  U_ z  e.  A  { <. x ,  y >.  |  ( x  e.  { z }  /\  y  e.  ( F `  z
) ) }
9 iunopab 4629 . . 3  |-  U_ z  e.  A  { <. x ,  y >.  |  ( x  e.  { z }  /\  y  e.  ( F `  z
) ) }  =  { <. x ,  y
>.  |  E. z  e.  A  ( x  e.  { z }  /\  y  e.  ( F `  z ) ) }
10 elsn 3896 . . . . . . . 8  |-  ( x  e.  { z }  <-> 
x  =  z )
11 equcom 1732 . . . . . . . 8  |-  ( x  =  z  <->  z  =  x )
1210, 11bitri 249 . . . . . . 7  |-  ( x  e.  { z }  <-> 
z  =  x )
1312anbi1i 695 . . . . . 6  |-  ( ( x  e.  { z }  /\  y  e.  ( F `  z
) )  <->  ( z  =  x  /\  y  e.  ( F `  z
) ) )
1413rexbii 2745 . . . . 5  |-  ( E. z  e.  A  ( x  e.  { z }  /\  y  e.  ( F `  z
) )  <->  E. z  e.  A  ( z  =  x  /\  y  e.  ( F `  z
) ) )
15 fveq2 5696 . . . . . . 7  |-  ( z  =  x  ->  ( F `  z )  =  ( F `  x ) )
1615eleq2d 2510 . . . . . 6  |-  ( z  =  x  ->  (
y  e.  ( F `
 z )  <->  y  e.  ( F `  x ) ) )
1716ceqsrexbv 3099 . . . . 5  |-  ( E. z  e.  A  ( z  =  x  /\  y  e.  ( F `  z ) )  <->  ( x  e.  A  /\  y  e.  ( F `  x
) ) )
1814, 17bitri 249 . . . 4  |-  ( E. z  e.  A  ( x  e.  { z }  /\  y  e.  ( F `  z
) )  <->  ( x  e.  A  /\  y  e.  ( F `  x
) ) )
1918opabbii 4361 . . 3  |-  { <. x ,  y >.  |  E. z  e.  A  (
x  e.  { z }  /\  y  e.  ( F `  z
) ) }  =  { <. x ,  y
>.  |  ( x  e.  A  /\  y  e.  ( F `  x
) ) }
208, 9, 193eqtri 2467 . 2  |-  U_ z  e.  A  ( {
z }  X.  ( F `  z )
)  =  { <. x ,  y >.  |  ( x  e.  A  /\  y  e.  ( F `  x ) ) }
211, 5, 203eqtri 2467 1  |-  T  =  { <. x ,  y
>.  |  ( x  e.  A  /\  y  e.  ( F `  x
) ) }
Colors of variables: wff setvar class
Syntax hints:    /\ wa 369    = wceq 1369    e. wcel 1756   E.wrex 2721   {csn 3882   U_ciun 4176   {copab 4354    X. cxp 4843   ` cfv 5423
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4418  ax-nul 4426  ax-pr 4536
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-ral 2725  df-rex 2726  df-rab 2729  df-v 2979  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-nul 3643  df-if 3797  df-sn 3883  df-pr 3885  df-op 3889  df-uni 4097  df-iun 4178  df-br 4298  df-opab 4356  df-xp 4851  df-iota 5386  df-fv 5431
This theorem is referenced by:  marypha2lem3  7692  marypha2lem4  7693  eulerpartlemgu  26765
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