MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  marypha2lem2 Structured version   Unicode version

Theorem marypha2lem2 7892
Description: Lemma for marypha2 7895. 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 4037 . . . 4  |-  ( x  =  z  ->  { x }  =  { z } )
3 fveq2 5864 . . . 4  |-  ( x  =  z  ->  ( F `  x )  =  ( F `  z ) )
42, 3xpeq12d 5024 . . 3  |-  ( x  =  z  ->  ( { x }  X.  ( F `  x ) )  =  ( { z }  X.  ( F `  z )
) )
54cbviunv 4364 . 2  |-  U_ x  e.  A  ( {
x }  X.  ( F `  x )
)  =  U_ z  e.  A  ( {
z }  X.  ( F `  z )
)
6 df-xp 5005 . . . . 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 4344 . . 3  |-  U_ z  e.  A  ( {
z }  X.  ( F `  z )
)  =  U_ z  e.  A  { <. x ,  y >.  |  ( x  e.  { z }  /\  y  e.  ( F `  z
) ) }
9 iunopab 4783 . . 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 4041 . . . . . . . 8  |-  ( x  e.  { z }  <-> 
x  =  z )
11 equcom 1743 . . . . . . . 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 2965 . . . . 5  |-  ( E. z  e.  A  ( x  e.  { z }  /\  y  e.  ( F `  z
) )  <->  E. z  e.  A  ( z  =  x  /\  y  e.  ( F `  z
) ) )
15 fveq2 5864 . . . . . . 7  |-  ( z  =  x  ->  ( F `  z )  =  ( F `  x ) )
1615eleq2d 2537 . . . . . 6  |-  ( z  =  x  ->  (
y  e.  ( F `
 z )  <->  y  e.  ( F `  x ) ) )
1716ceqsrexbv 3238 . . . . 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 4511 . . 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 2500 . 2  |-  U_ z  e.  A  ( {
z }  X.  ( F `  z )
)  =  { <. x ,  y >.  |  ( x  e.  A  /\  y  e.  ( F `  x ) ) }
211, 5, 203eqtri 2500 1  |-  T  =  { <. x ,  y
>.  |  ( x  e.  A  /\  y  e.  ( F `  x
) ) }
Colors of variables: wff setvar class
Syntax hints:    /\ wa 369    = wceq 1379    e. wcel 1767   E.wrex 2815   {csn 4027   U_ciun 4325   {copab 4504    X. cxp 4997   ` cfv 5586
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-xp 5005  df-iota 5549  df-fv 5594
This theorem is referenced by:  marypha2lem3  7893  marypha2lem4  7894  eulerpartlemgu  27956
  Copyright terms: Public domain W3C validator