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Theorem prtlem19 31881
Description: Lemma for prter2 31884. (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 31880 . . . . 5  |-  ( Prt 
A  ->  ( (
v  e.  A  /\  z  e.  v )  ->  ( w  e.  v  <-> 
z  .~  w )
) )
32imp 427 . . . 4  |-  ( ( Prt  A  /\  (
v  e.  A  /\  z  e.  v )
)  ->  ( w  e.  v  <->  z  .~  w
) )
4 vex 3061 . . . . 5  |-  w  e. 
_V
5 vex 3061 . . . . 5  |-  z  e. 
_V
64, 5elec 7387 . . . 4  |-  ( w  e.  [ z ]  .~  <->  z  .~  w
)
73, 6syl6bbr 263 . . 3  |-  ( ( Prt  A  /\  (
v  e.  A  /\  z  e.  v )
)  ->  ( w  e.  v  <->  w  e.  [ z ]  .~  ) )
87eqrdv 2399 . 2  |-  ( ( Prt  A  /\  (
v  e.  A  /\  z  e.  v )
)  ->  v  =  [ z ]  .~  )
98ex 432 1  |-  ( Prt 
A  ->  ( (
v  e.  A  /\  z  e.  v )  ->  v  =  [ z ]  .~  ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1405    e. wcel 1842   E.wrex 2754   class class class wbr 4394   {copab 4451   [cec 7345   Prt wprt 31874
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4516  ax-nul 4524  ax-pr 4629
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-rab 2762  df-v 3060  df-sbc 3277  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-sn 3972  df-pr 3974  df-op 3978  df-br 4395  df-opab 4453  df-xp 4828  df-cnv 4830  df-dm 4832  df-rn 4833  df-res 4834  df-ima 4835  df-ec 7349  df-prt 31875
This theorem is referenced by:  prter2  31884
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