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Theorem prtlem19 30210
Description: Lemma for prter2 30213. (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 30209 . . . . 5  |-  ( Prt 
A  ->  ( (
v  e.  A  /\  z  e.  v )  ->  ( w  e.  v  <-> 
z  .~  w )
) )
32imp 429 . . . 4  |-  ( ( Prt  A  /\  (
v  e.  A  /\  z  e.  v )
)  ->  ( w  e.  v  <->  z  .~  w
) )
4 vex 3109 . . . . 5  |-  w  e. 
_V
5 vex 3109 . . . . 5  |-  z  e. 
_V
64, 5elec 7341 . . . 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 2457 . 2  |-  ( ( Prt  A  /\  (
v  e.  A  /\  z  e.  v )
)  ->  v  =  [ z ]  .~  )
98ex 434 1  |-  ( Prt 
A  ->  ( (
v  e.  A  /\  z  e.  v )  ->  v  =  [ z ]  .~  ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1374    e. wcel 1762   E.wrex 2808   class class class wbr 4440   {copab 4497   [cec 7299   Prt wprt 30203
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-sep 4561  ax-nul 4569  ax-pr 4679
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3108  df-sbc 3325  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-sn 4021  df-pr 4023  df-op 4027  df-br 4441  df-opab 4499  df-xp 4998  df-cnv 5000  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-ec 7303  df-prt 30204
This theorem is referenced by:  prter2  30213
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