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Theorem prtlem16 30578
Description: Lemma for prtex 30589, prter2 30590 and prter3 30591. (Contributed by Rodolfo Medina, 14-Oct-2010.) (Revised by Mario Carneiro, 12-Aug-2015.)
Hypothesis
Ref Expression
prtlem13.1  |-  .~  =  { <. x ,  y
>.  |  E. u  e.  A  ( x  e.  u  /\  y  e.  u ) }
Assertion
Ref Expression
prtlem16  |-  dom  .~  =  U. A
Distinct variable group:    x, u, y, A
Allowed substitution hints:    .~ ( x, y, u)

Proof of Theorem prtlem16
Dummy variables  v  w  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 vex 3096 . . . 4  |-  z  e. 
_V
21eldm 5186 . . 3  |-  ( z  e.  dom  .~  <->  E. w  z  .~  w )
3 prtlem13.1 . . . . 5  |-  .~  =  { <. x ,  y
>.  |  E. u  e.  A  ( x  e.  u  /\  y  e.  u ) }
43prtlem13 30577 . . . 4  |-  ( z  .~  w  <->  E. v  e.  A  ( z  e.  v  /\  w  e.  v ) )
54exbii 1652 . . 3  |-  ( E. w  z  .~  w  <->  E. w E. v  e.  A  ( z  e.  v  /\  w  e.  v ) )
6 elunii 4235 . . . . . . . 8  |-  ( ( z  e.  v  /\  v  e.  A )  ->  z  e.  U. A
)
76ancoms 453 . . . . . . 7  |-  ( ( v  e.  A  /\  z  e.  v )  ->  z  e.  U. A
)
87adantrr 716 . . . . . 6  |-  ( ( v  e.  A  /\  ( z  e.  v  /\  w  e.  v ) )  ->  z  e.  U. A )
98rexlimiva 2929 . . . . 5  |-  ( E. v  e.  A  ( z  e.  v  /\  w  e.  v )  ->  z  e.  U. A
)
109exlimiv 1707 . . . 4  |-  ( E. w E. v  e.  A  ( z  e.  v  /\  w  e.  v )  ->  z  e.  U. A )
11 eluni2 4234 . . . . 5  |-  ( z  e.  U. A  <->  E. v  e.  A  z  e.  v )
12 eleq1 2513 . . . . . . . . 9  |-  ( w  =  z  ->  (
w  e.  v  <->  z  e.  v ) )
1312anbi2d 703 . . . . . . . 8  |-  ( w  =  z  ->  (
( z  e.  v  /\  w  e.  v )  <->  ( z  e.  v  /\  z  e.  v ) ) )
14 pm4.24 643 . . . . . . . 8  |-  ( z  e.  v  <->  ( z  e.  v  /\  z  e.  v ) )
1513, 14syl6bbr 263 . . . . . . 7  |-  ( w  =  z  ->  (
( z  e.  v  /\  w  e.  v )  <->  z  e.  v ) )
1615rexbidv 2952 . . . . . 6  |-  ( w  =  z  ->  ( E. v  e.  A  ( z  e.  v  /\  w  e.  v )  <->  E. v  e.  A  z  e.  v )
)
171, 16spcev 3185 . . . . 5  |-  ( E. v  e.  A  z  e.  v  ->  E. w E. v  e.  A  ( z  e.  v  /\  w  e.  v ) )
1811, 17sylbi 195 . . . 4  |-  ( z  e.  U. A  ->  E. w E. v  e.  A  ( z  e.  v  /\  w  e.  v ) )
1910, 18impbii 188 . . 3  |-  ( E. w E. v  e.  A  ( z  e.  v  /\  w  e.  v )  <->  z  e.  U. A )
202, 5, 193bitri 271 . 2  |-  ( z  e.  dom  .~  <->  z  e.  U. A )
2120eqriv 2437 1  |-  dom  .~  =  U. A
Colors of variables: wff setvar class
Syntax hints:    /\ wa 369    = wceq 1381   E.wex 1597    e. wcel 1802   E.wrex 2792   U.cuni 4230   class class class wbr 4433   {copab 4490   dom cdm 4985
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-sep 4554  ax-nul 4562  ax-pr 4672
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-ral 2796  df-rex 2797  df-rab 2800  df-v 3095  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-nul 3768  df-if 3923  df-sn 4011  df-pr 4013  df-op 4017  df-uni 4231  df-br 4434  df-opab 4492  df-dm 4995
This theorem is referenced by:  prtlem400  30579  prter1  30588
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