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Theorem prtlem16 26718
Description: Lemma for prtex 26729, prter2 26730 and prter3 26731. (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 2959 . . . 4  |-  z  e. 
_V
21eldm 5067 . . 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 26717 . . . 4  |-  ( z  .~  w  <->  E. v  e.  A  ( z  e.  v  /\  w  e.  v ) )
54exbii 1592 . . 3  |-  ( E. w  z  .~  w  <->  E. w E. v  e.  A  ( z  e.  v  /\  w  e.  v ) )
6 elunii 4020 . . . . . . . 8  |-  ( ( z  e.  v  /\  v  e.  A )  ->  z  e.  U. A
)
76ancoms 440 . . . . . . 7  |-  ( ( v  e.  A  /\  z  e.  v )  ->  z  e.  U. A
)
87adantrr 698 . . . . . 6  |-  ( ( v  e.  A  /\  ( z  e.  v  /\  w  e.  v ) )  ->  z  e.  U. A )
98rexlimiva 2825 . . . . 5  |-  ( E. v  e.  A  ( z  e.  v  /\  w  e.  v )  ->  z  e.  U. A
)
109exlimiv 1644 . . . 4  |-  ( E. w E. v  e.  A  ( z  e.  v  /\  w  e.  v )  ->  z  e.  U. A )
11 eluni2 4019 . . . . 5  |-  ( z  e.  U. A  <->  E. v  e.  A  z  e.  v )
12 eleq1 2496 . . . . . . . . 9  |-  ( w  =  z  ->  (
w  e.  v  <->  z  e.  v ) )
1312anbi2d 685 . . . . . . . 8  |-  ( w  =  z  ->  (
( z  e.  v  /\  w  e.  v )  <->  ( z  e.  v  /\  z  e.  v ) ) )
14 pm4.24 625 . . . . . . . 8  |-  ( z  e.  v  <->  ( z  e.  v  /\  z  e.  v ) )
1513, 14syl6bbr 255 . . . . . . 7  |-  ( w  =  z  ->  (
( z  e.  v  /\  w  e.  v )  <->  z  e.  v ) )
1615rexbidv 2726 . . . . . 6  |-  ( w  =  z  ->  ( E. v  e.  A  ( z  e.  v  /\  w  e.  v )  <->  E. v  e.  A  z  e.  v )
)
171, 16spcev 3043 . . . . 5  |-  ( E. v  e.  A  z  e.  v  ->  E. w E. v  e.  A  ( z  e.  v  /\  w  e.  v ) )
1811, 17sylbi 188 . . . 4  |-  ( z  e.  U. A  ->  E. w E. v  e.  A  ( z  e.  v  /\  w  e.  v ) )
1910, 18impbii 181 . . 3  |-  ( E. w E. v  e.  A  ( z  e.  v  /\  w  e.  v )  <->  z  e.  U. A )
202, 5, 193bitri 263 . 2  |-  ( z  e.  dom  .~  <->  z  e.  U. A )
2120eqriv 2433 1  |-  dom  .~  =  U. A
Colors of variables: wff set class
Syntax hints:    /\ wa 359   E.wex 1550    = wceq 1652    e. wcel 1725   E.wrex 2706   U.cuni 4015   class class class wbr 4212   {copab 4265   dom cdm 4878
This theorem is referenced by:  prtlem400  26719  prter1  26728
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pr 4403
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-dm 4888
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