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Theorem prtlem19 26746
Description: Lemma for prter2 26749. (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 26745 . . . . 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 2791 . . . . 5  |-  w  e. 
_V
5 vex 2791 . . . . 5  |-  z  e. 
_V
64, 5elec 6699 . . . 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 2281 . 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 1623    e. wcel 1684   E.wrex 2544   class class class wbr 4023   {copab 4076   [cec 6658   Prt wprt 26739
This theorem is referenced by:  prter2  26749
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-opab 4078  df-xp 4695  df-cnv 4697  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-ec 6662  df-prt 26740
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