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Theorem prtlem16 26737
Description: Lemma for prtex 26748, prter2 26749 and prter3 26750. (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 2791 . . . 4  |-  z  e. 
_V
21eldm 4876 . . 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 26736 . . . 4  |-  ( z  .~  w  <->  E. v  e.  A  ( z  e.  v  /\  w  e.  v ) )
54exbii 1569 . . 3  |-  ( E. w  z  .~  w  <->  E. w E. v  e.  A  ( z  e.  v  /\  w  e.  v ) )
6 elunii 3832 . . . . . . . 8  |-  ( ( z  e.  v  /\  v  e.  A )  ->  z  e.  U. A
)
76ancoms 439 . . . . . . 7  |-  ( ( v  e.  A  /\  z  e.  v )  ->  z  e.  U. A
)
87adantrr 697 . . . . . 6  |-  ( ( v  e.  A  /\  ( z  e.  v  /\  w  e.  v ) )  ->  z  e.  U. A )
98rexlimiva 2662 . . . . 5  |-  ( E. v  e.  A  ( z  e.  v  /\  w  e.  v )  ->  z  e.  U. A
)
109exlimiv 1666 . . . 4  |-  ( E. w E. v  e.  A  ( z  e.  v  /\  w  e.  v )  ->  z  e.  U. A )
11 eluni2 3831 . . . . 5  |-  ( z  e.  U. A  <->  E. v  e.  A  z  e.  v )
12 eleq1 2343 . . . . . . . . 9  |-  ( w  =  z  ->  (
w  e.  v  <->  z  e.  v ) )
1312anbi2d 684 . . . . . . . 8  |-  ( w  =  z  ->  (
( z  e.  v  /\  w  e.  v )  <->  ( z  e.  v  /\  z  e.  v ) ) )
14 pm4.24 624 . . . . . . . 8  |-  ( z  e.  v  <->  ( z  e.  v  /\  z  e.  v ) )
1513, 14syl6bbr 254 . . . . . . 7  |-  ( w  =  z  ->  (
( z  e.  v  /\  w  e.  v )  <->  z  e.  v ) )
1615rexbidv 2564 . . . . . 6  |-  ( w  =  z  ->  ( E. v  e.  A  ( z  e.  v  /\  w  e.  v )  <->  E. v  e.  A  z  e.  v )
)
171, 16spcev 2875 . . . . 5  |-  ( E. v  e.  A  z  e.  v  ->  E. w E. v  e.  A  ( z  e.  v  /\  w  e.  v ) )
1811, 17sylbi 187 . . . 4  |-  ( z  e.  U. A  ->  E. w E. v  e.  A  ( z  e.  v  /\  w  e.  v ) )
1910, 18impbii 180 . . 3  |-  ( E. w E. v  e.  A  ( z  e.  v  /\  w  e.  v )  <->  z  e.  U. A )
202, 5, 193bitri 262 . 2  |-  ( z  e.  dom  .~  <->  z  e.  U. A )
2120eqriv 2280 1  |-  dom  .~  =  U. A
Colors of variables: wff set class
Syntax hints:    /\ wa 358   E.wex 1528    = wceq 1623    e. wcel 1684   E.wrex 2544   U.cuni 3827   class class class wbr 4023   {copab 4076   dom cdm 4689
This theorem is referenced by:  prtlem400  26738  prter1  26747
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-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-uni 3828  df-br 4024  df-opab 4078  df-dm 4699
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