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Theorem prter1 26747
Description: Every partition generates an equivalence relation. (Contributed by Rodolfo Medina, 13-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
prter1  |-  ( Prt 
A  ->  .~  Er  U. A )
Distinct variable group:    x, u, y, A
Allowed substitution hints:    .~ ( x, y, u)

Proof of Theorem prter1
Dummy variables  q  p  r  v  w  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 prtlem18.1 . . . 4  |-  .~  =  { <. x ,  y
>.  |  E. u  e.  A  ( x  e.  u  /\  y  e.  u ) }
21relopabi 4811 . . 3  |-  Rel  .~
32a1i 10 . 2  |-  ( Prt 
A  ->  Rel  .~  )
41prtlem16 26737 . . 3  |-  dom  .~  =  U. A
54a1i 10 . 2  |-  ( Prt 
A  ->  dom  .~  =  U. A )
6 prtlem15 26743 . . . . . 6  |-  ( Prt 
A  ->  ( E. v  e.  A  E. q  e.  A  (
( z  e.  v  /\  w  e.  v )  /\  ( w  e.  q  /\  p  e.  q ) )  ->  E. r  e.  A  ( z  e.  r  /\  p  e.  r ) ) )
71prtlem13 26736 . . . . . . . 8  |-  ( z  .~  w  <->  E. v  e.  A  ( z  e.  v  /\  w  e.  v ) )
81prtlem13 26736 . . . . . . . 8  |-  ( w  .~  p  <->  E. q  e.  A  ( w  e.  q  /\  p  e.  q ) )
97, 8anbi12i 678 . . . . . . 7  |-  ( ( z  .~  w  /\  w  .~  p )  <->  ( E. v  e.  A  (
z  e.  v  /\  w  e.  v )  /\  E. q  e.  A  ( w  e.  q  /\  p  e.  q
) ) )
10 reeanv 2707 . . . . . . 7  |-  ( E. v  e.  A  E. q  e.  A  (
( z  e.  v  /\  w  e.  v )  /\  ( w  e.  q  /\  p  e.  q ) )  <->  ( E. v  e.  A  (
z  e.  v  /\  w  e.  v )  /\  E. q  e.  A  ( w  e.  q  /\  p  e.  q
) ) )
119, 10bitr4i 243 . . . . . 6  |-  ( ( z  .~  w  /\  w  .~  p )  <->  E. v  e.  A  E. q  e.  A  ( (
z  e.  v  /\  w  e.  v )  /\  ( w  e.  q  /\  p  e.  q ) ) )
121prtlem13 26736 . . . . . 6  |-  ( z  .~  p  <->  E. r  e.  A  ( z  e.  r  /\  p  e.  r ) )
136, 11, 123imtr4g 261 . . . . 5  |-  ( Prt 
A  ->  ( (
z  .~  w  /\  w  .~  p )  -> 
z  .~  p )
)
14 pm3.22 436 . . . . . . 7  |-  ( ( z  e.  v  /\  w  e.  v )  ->  ( w  e.  v  /\  z  e.  v ) )
1514reximi 2650 . . . . . 6  |-  ( E. v  e.  A  ( z  e.  v  /\  w  e.  v )  ->  E. v  e.  A  ( w  e.  v  /\  z  e.  v
) )
161prtlem13 26736 . . . . . 6  |-  ( w  .~  z  <->  E. v  e.  A  ( w  e.  v  /\  z  e.  v ) )
1715, 7, 163imtr4i 257 . . . . 5  |-  ( z  .~  w  ->  w  .~  z )
1813, 17jctil 523 . . . 4  |-  ( Prt 
A  ->  ( (
z  .~  w  ->  w  .~  z )  /\  ( ( z  .~  w  /\  w  .~  p
)  ->  z  .~  p ) ) )
1918alrimivv 1618 . . 3  |-  ( Prt 
A  ->  A. w A. p ( ( z  .~  w  ->  w  .~  z )  /\  (
( z  .~  w  /\  w  .~  p
)  ->  z  .~  p ) ) )
2019alrimiv 1617 . 2  |-  ( Prt 
A  ->  A. z A. w A. p ( ( z  .~  w  ->  w  .~  z )  /\  ( ( z  .~  w  /\  w  .~  p )  ->  z  .~  p ) ) )
21 dfer2 6661 . 2  |-  (  .~  Er  U. A  <->  ( Rel  .~ 
/\  dom  .~  =  U. A  /\  A. z A. w A. p ( ( z  .~  w  ->  w  .~  z )  /\  ( ( z  .~  w  /\  w  .~  p )  ->  z  .~  p ) ) ) )
223, 5, 20, 21syl3anbrc 1136 1  |-  ( Prt 
A  ->  .~  Er  U. A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358   A.wal 1527    = wceq 1623    e. wcel 1684   E.wrex 2544   U.cuni 3827   class class class wbr 4023   {copab 4076   dom cdm 4689   Rel wrel 4694    Er wer 6657   Prt wprt 26739
This theorem is referenced by:  prtex  26748
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-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-er 6660  df-prt 26740
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