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Theorem prter1 26709
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 4992 . . 3  |-  Rel  .~
32a1i 11 . 2  |-  ( Prt 
A  ->  Rel  .~  )
41prtlem16 26699 . . 3  |-  dom  .~  =  U. A
54a1i 11 . 2  |-  ( Prt 
A  ->  dom  .~  =  U. A )
6 prtlem15 26705 . . . . . 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 26698 . . . . . . . 8  |-  ( z  .~  w  <->  E. v  e.  A  ( z  e.  v  /\  w  e.  v ) )
81prtlem13 26698 . . . . . . . 8  |-  ( w  .~  p  <->  E. q  e.  A  ( w  e.  q  /\  p  e.  q ) )
97, 8anbi12i 679 . . . . . . 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 2867 . . . . . . 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 244 . . . . . 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 26698 . . . . . 6  |-  ( z  .~  p  <->  E. r  e.  A  ( z  e.  r  /\  p  e.  r ) )
136, 11, 123imtr4g 262 . . . . 5  |-  ( Prt 
A  ->  ( (
z  .~  w  /\  w  .~  p )  -> 
z  .~  p )
)
14 pm3.22 437 . . . . . . 7  |-  ( ( z  e.  v  /\  w  e.  v )  ->  ( w  e.  v  /\  z  e.  v ) )
1514reximi 2805 . . . . . 6  |-  ( E. v  e.  A  ( z  e.  v  /\  w  e.  v )  ->  E. v  e.  A  ( w  e.  v  /\  z  e.  v
) )
161prtlem13 26698 . . . . . 6  |-  ( w  .~  z  <->  E. v  e.  A  ( w  e.  v  /\  z  e.  v ) )
1715, 7, 163imtr4i 258 . . . . 5  |-  ( z  .~  w  ->  w  .~  z )
1813, 17jctil 524 . . . 4  |-  ( Prt 
A  ->  ( (
z  .~  w  ->  w  .~  z )  /\  ( ( z  .~  w  /\  w  .~  p
)  ->  z  .~  p ) ) )
1918alrimivv 1642 . . 3  |-  ( Prt 
A  ->  A. w A. p ( ( z  .~  w  ->  w  .~  z )  /\  (
( z  .~  w  /\  w  .~  p
)  ->  z  .~  p ) ) )
2019alrimiv 1641 . 2  |-  ( Prt 
A  ->  A. z A. w A. p ( ( z  .~  w  ->  w  .~  z )  /\  ( ( z  .~  w  /\  w  .~  p )  ->  z  .~  p ) ) )
21 dfer2 6898 . 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 1138 1  |-  ( Prt 
A  ->  .~  Er  U. A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359   A.wal 1549    = wceq 1652   E.wrex 2698   U.cuni 4007   class class class wbr 4204   {copab 4257   dom cdm 4870   Rel wrel 4875    Er wer 6894   Prt wprt 26701
This theorem is referenced by:  prtex  26710
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 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395
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 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-er 6897  df-prt 26702
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