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Theorem marypha2lem2 7441
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
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 3825 . . . 4  |-  ( x  =  z  ->  { x }  =  { z } )
3 fveq2 5728 . . . 4  |-  ( x  =  z  ->  ( F `  x )  =  ( F `  z ) )
42, 3xpeq12d 4903 . . 3  |-  ( x  =  z  ->  ( { x }  X.  ( F `  x ) )  =  ( { z }  X.  ( F `  z )
) )
54cbviunv 4130 . 2  |-  U_ x  e.  A  ( {
x }  X.  ( F `  x )
)  =  U_ z  e.  A  ( {
z }  X.  ( F `  z )
)
6 df-xp 4884 . . . . 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 4111 . . 3  |-  U_ z  e.  A  ( {
z }  X.  ( F `  z )
)  =  U_ z  e.  A  { <. x ,  y >.  |  ( x  e.  { z }  /\  y  e.  ( F `  z
) ) }
9 iunopab 4486 . . 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 3829 . . . . . . . 8  |-  ( x  e.  { z }  <-> 
x  =  z )
11 equcom 1692 . . . . . . . 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 2730 . . . . 5  |-  ( E. z  e.  A  ( x  e.  { z }  /\  y  e.  ( F `  z
) )  <->  E. z  e.  A  ( z  =  x  /\  y  e.  ( F `  z
) ) )
15 fveq2 5728 . . . . . . 7  |-  ( z  =  x  ->  ( F `  z )  =  ( F `  x ) )
1615eleq2d 2503 . . . . . 6  |-  ( z  =  x  ->  (
y  e.  ( F `
 z )  <->  y  e.  ( F `  x ) ) )
1716ceqsrexbv 3070 . . . . 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 4272 . . 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 2460 . 2  |-  U_ z  e.  A  ( {
z }  X.  ( F `  z )
)  =  { <. x ,  y >.  |  ( x  e.  A  /\  y  e.  ( F `  x ) ) }
211, 5, 203eqtri 2460 1  |-  T  =  { <. x ,  y
>.  |  ( x  e.  A  /\  y  e.  ( F `  x
) ) }
Colors of variables: wff set class
Syntax hints:    /\ wa 359    = wceq 1652    e. wcel 1725   E.wrex 2706   {csn 3814   U_ciun 4093   {copab 4265    X. cxp 4876   ` cfv 5454
This theorem is referenced by:  marypha2lem3  7442  marypha2lem4  7443
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-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  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-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-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-iota 5418  df-fv 5462
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