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Theorem prtlem19 26849
Description: Lemma for prter2 26852. (Contributed by Rodolfo Medina, 15-Oct-2010.) (Revised by Mario Carneiro, 12-Aug-2015.)
Hypothesis
Ref Expression
prtlem18.1  |-  .~  =  { <. x ,  y
>.  |  E. u  e.  A  ( x  e.  u  /\  y  e.  u ) }
Assertion
Ref Expression
prtlem19  |-  ( Prt 
A  ->  ( (
v  e.  A  /\  z  e.  v )  ->  v  =  [ z ]  .~  ) )
Distinct variable groups:    v, u, x, y, z, A    v,  .~ , z
Allowed substitution hints:    .~ ( x, y, u)

Proof of Theorem prtlem19
Dummy variable  w is distinct from all other variables.
StepHypRef Expression
1 prtlem18.1 . . . . . 6  |-  .~  =  { <. x ,  y
>.  |  E. u  e.  A  ( x  e.  u  /\  y  e.  u ) }
21prtlem18 26848 . . . . 5  |-  ( Prt 
A  ->  ( (
v  e.  A  /\  z  e.  v )  ->  ( w  e.  v  <-> 
z  .~  w )
) )
32imp 418 . . . 4  |-  ( ( Prt  A  /\  (
v  e.  A  /\  z  e.  v )
)  ->  ( w  e.  v  <->  z  .~  w
) )
4 vex 2804 . . . . 5  |-  w  e. 
_V
5 vex 2804 . . . . 5  |-  z  e. 
_V
64, 5elec 6715 . . . 4  |-  ( w  e.  [ z ]  .~  <->  z  .~  w
)
73, 6syl6bbr 254 . . 3  |-  ( ( Prt  A  /\  (
v  e.  A  /\  z  e.  v )
)  ->  ( w  e.  v  <->  w  e.  [ z ]  .~  ) )
87eqrdv 2294 . 2  |-  ( ( Prt  A  /\  (
v  e.  A  /\  z  e.  v )
)  ->  v  =  [ z ]  .~  )
98ex 423 1  |-  ( Prt 
A  ->  ( (
v  e.  A  /\  z  e.  v )  ->  v  =  [ z ]  .~  ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696   E.wrex 2557   class class class wbr 4039   {copab 4092   [cec 6674   Prt wprt 26842
This theorem is referenced by:  prter2  26852
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-eu 2160  df-mo 2161  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-sbc 3005  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-br 4040  df-opab 4094  df-xp 4711  df-cnv 4713  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-ec 6678  df-prt 26843
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