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Theorem prtlem16 26840
Description: Lemma for prtex 26851, prter2 26852 and prter3 26853. (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 2804 . . . 4  |-  z  e. 
_V
21eldm 4892 . . 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 26839 . . . 4  |-  ( z  .~  w  <->  E. v  e.  A  ( z  e.  v  /\  w  e.  v ) )
54exbii 1572 . . 3  |-  ( E. w  z  .~  w  <->  E. w E. v  e.  A  ( z  e.  v  /\  w  e.  v ) )
6 elunii 3848 . . . . . . . 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 2675 . . . . 5  |-  ( E. v  e.  A  ( z  e.  v  /\  w  e.  v )  ->  z  e.  U. A
)
109exlimiv 1624 . . . 4  |-  ( E. w E. v  e.  A  ( z  e.  v  /\  w  e.  v )  ->  z  e.  U. A )
11 eluni2 3847 . . . . 5  |-  ( z  e.  U. A  <->  E. v  e.  A  z  e.  v )
12 eleq1 2356 . . . . . . . . 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 2577 . . . . . 6  |-  ( w  =  z  ->  ( E. v  e.  A  ( z  e.  v  /\  w  e.  v )  <->  E. v  e.  A  z  e.  v )
)
171, 16spcev 2888 . . . . 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 2293 1  |-  dom  .~  =  U. A
Colors of variables: wff set class
Syntax hints:    /\ wa 358   E.wex 1531    = wceq 1632    e. wcel 1696   E.wrex 2557   U.cuni 3843   class class class wbr 4039   {copab 4092   dom cdm 4705
This theorem is referenced by:  prtlem400  26841  prter1  26850
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-dm 4715
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