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Theorem marypha2lem2 7399
Description: Lemma for marypha2 7402. 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 3785 . . . 4  |-  ( x  =  z  ->  { x }  =  { z } )
3 fveq2 5687 . . . 4  |-  ( x  =  z  ->  ( F `  x )  =  ( F `  z ) )
42, 3xpeq12d 4862 . . 3  |-  ( x  =  z  ->  ( { x }  X.  ( F `  x ) )  =  ( { z }  X.  ( F `  z )
) )
54cbviunv 4090 . 2  |-  U_ x  e.  A  ( {
x }  X.  ( F `  x )
)  =  U_ z  e.  A  ( {
z }  X.  ( F `  z )
)
6 df-xp 4843 . . . . 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 4071 . . 3  |-  U_ z  e.  A  ( {
z }  X.  ( F `  z )
)  =  U_ z  e.  A  { <. x ,  y >.  |  ( x  e.  { z }  /\  y  e.  ( F `  z
) ) }
9 iunopab 4446 . . 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 3789 . . . . . . . 8  |-  ( x  e.  { z }  <-> 
x  =  z )
11 equcom 1688 . . . . . . . 8  |-  ( x  =  z  <->  z  =  x )
1210, 11bitri 241 . . . . . . 7  |-  ( x  e.  { z }  <-> 
z  =  x )
1312anbi1i 677 . . . . . 6  |-  ( ( x  e.  { z }  /\  y  e.  ( F `  z
) )  <->  ( z  =  x  /\  y  e.  ( F `  z
) ) )
1413rexbii 2691 . . . . 5  |-  ( E. z  e.  A  ( x  e.  { z }  /\  y  e.  ( F `  z
) )  <->  E. z  e.  A  ( z  =  x  /\  y  e.  ( F `  z
) ) )
15 fveq2 5687 . . . . . . 7  |-  ( z  =  x  ->  ( F `  z )  =  ( F `  x ) )
1615eleq2d 2471 . . . . . 6  |-  ( z  =  x  ->  (
y  e.  ( F `
 z )  <->  y  e.  ( F `  x ) ) )
1716ceqsrexbv 3030 . . . . 5  |-  ( E. z  e.  A  ( z  =  x  /\  y  e.  ( F `  z ) )  <->  ( x  e.  A  /\  y  e.  ( F `  x
) ) )
1814, 17bitri 241 . . . 4  |-  ( E. z  e.  A  ( x  e.  { z }  /\  y  e.  ( F `  z
) )  <->  ( x  e.  A  /\  y  e.  ( F `  x
) ) )
1918opabbii 4232 . . 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 2428 . 2  |-  U_ z  e.  A  ( {
z }  X.  ( F `  z )
)  =  { <. x ,  y >.  |  ( x  e.  A  /\  y  e.  ( F `  x ) ) }
211, 5, 203eqtri 2428 1  |-  T  =  { <. x ,  y
>.  |  ( x  e.  A  /\  y  e.  ( F `  x
) ) }
Colors of variables: wff set class
Syntax hints:    /\ wa 359    = wceq 1649    e. wcel 1721   E.wrex 2667   {csn 3774   U_ciun 4053   {copab 4225    X. cxp 4835   ` cfv 5413
This theorem is referenced by:  marypha2lem3  7400  marypha2lem4  7401
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pr 4363
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-xp 4843  df-iota 5377  df-fv 5421
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