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Theorem marypha2lem2 7205
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
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 3664 . . . 4  |-  ( x  =  z  ->  { x }  =  { z } )
3 fveq2 5541 . . . 4  |-  ( x  =  z  ->  ( F `  x )  =  ( F `  z ) )
42, 3xpeq12d 4730 . . 3  |-  ( x  =  z  ->  ( { x }  X.  ( F `  x ) )  =  ( { z }  X.  ( F `  z )
) )
54cbviunv 3957 . 2  |-  U_ x  e.  A  ( {
x }  X.  ( F `  x )
)  =  U_ z  e.  A  ( {
z }  X.  ( F `  z )
)
6 df-xp 4711 . . . . 5  |-  ( { z }  X.  ( F `  z )
)  =  { <. x ,  y >.  |  ( x  e.  { z }  /\  y  e.  ( F `  z
) ) }
76a1i 10 . . . 4  |-  ( z  e.  A  ->  ( { z }  X.  ( F `  z ) )  =  { <. x ,  y >.  |  ( x  e.  { z }  /\  y  e.  ( F `  z
) ) } )
87iuneq2i 3939 . . 3  |-  U_ z  e.  A  ( {
z }  X.  ( F `  z )
)  =  U_ z  e.  A  { <. x ,  y >.  |  ( x  e.  { z }  /\  y  e.  ( F `  z
) ) }
9 iunopab 4312 . . 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 3668 . . . . . . . 8  |-  ( x  e.  { z }  <-> 
x  =  z )
11 equcom 1665 . . . . . . . 8  |-  ( x  =  z  <->  z  =  x )
1210, 11bitri 240 . . . . . . 7  |-  ( x  e.  { z }  <-> 
z  =  x )
1312anbi1i 676 . . . . . 6  |-  ( ( x  e.  { z }  /\  y  e.  ( F `  z
) )  <->  ( z  =  x  /\  y  e.  ( F `  z
) ) )
1413rexbii 2581 . . . . 5  |-  ( E. z  e.  A  ( x  e.  { z }  /\  y  e.  ( F `  z
) )  <->  E. z  e.  A  ( z  =  x  /\  y  e.  ( F `  z
) ) )
15 fveq2 5541 . . . . . . 7  |-  ( z  =  x  ->  ( F `  z )  =  ( F `  x ) )
1615eleq2d 2363 . . . . . 6  |-  ( z  =  x  ->  (
y  e.  ( F `
 z )  <->  y  e.  ( F `  x ) ) )
1716ceqsrexbv 2915 . . . . 5  |-  ( E. z  e.  A  ( z  =  x  /\  y  e.  ( F `  z ) )  <->  ( x  e.  A  /\  y  e.  ( F `  x
) ) )
1814, 17bitri 240 . . . 4  |-  ( E. z  e.  A  ( x  e.  { z }  /\  y  e.  ( F `  z
) )  <->  ( x  e.  A  /\  y  e.  ( F `  x
) ) )
1918opabbii 4099 . . 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 2320 . 2  |-  U_ z  e.  A  ( {
z }  X.  ( F `  z )
)  =  { <. x ,  y >.  |  ( x  e.  A  /\  y  e.  ( F `  x ) ) }
211, 5, 203eqtri 2320 1  |-  T  =  { <. x ,  y
>.  |  ( x  e.  A  /\  y  e.  ( F `  x
) ) }
Colors of variables: wff set class
Syntax hints:    /\ wa 358    = wceq 1632    e. wcel 1696   E.wrex 2557   {csn 3653   U_ciun 3921   {copab 4092    X. cxp 4703   ` cfv 5271
This theorem is referenced by:  marypha2lem3  7206  marypha2lem4  7207
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-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  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-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-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-iota 5235  df-fv 5279
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