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Theorem prtlem19 26718
Description: Lemma for prter2 26721. (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 26717 . . . . 5  |-  ( Prt 
A  ->  ( (
v  e.  A  /\  z  e.  v )  ->  ( w  e.  v  <-> 
z  .~  w )
) )
32imp 419 . . . 4  |-  ( ( Prt  A  /\  (
v  e.  A  /\  z  e.  v )
)  ->  ( w  e.  v  <->  z  .~  w
) )
4 vex 2951 . . . . 5  |-  w  e. 
_V
5 vex 2951 . . . . 5  |-  z  e. 
_V
64, 5elec 6936 . . . 4  |-  ( w  e.  [ z ]  .~  <->  z  .~  w
)
73, 6syl6bbr 255 . . 3  |-  ( ( Prt  A  /\  (
v  e.  A  /\  z  e.  v )
)  ->  ( w  e.  v  <->  w  e.  [ z ]  .~  ) )
87eqrdv 2433 . 2  |-  ( ( Prt  A  /\  (
v  e.  A  /\  z  e.  v )
)  ->  v  =  [ z ]  .~  )
98ex 424 1  |-  ( Prt 
A  ->  ( (
v  e.  A  /\  z  e.  v )  ->  v  =  [ z ]  .~  ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725   E.wrex 2698   class class class wbr 4204   {copab 4257   [cec 6895   Prt wprt 26711
This theorem is referenced by:  prter2  26721
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 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395
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-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-br 4205  df-opab 4259  df-xp 4876  df-cnv 4878  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-ec 6899  df-prt 26712
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