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Theorem prter1 30554
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 5134 . . 3  |-  Rel  .~
32a1i 11 . 2  |-  ( Prt 
A  ->  Rel  .~  )
41prtlem16 30544 . . 3  |-  dom  .~  =  U. A
54a1i 11 . 2  |-  ( Prt 
A  ->  dom  .~  =  U. A )
6 prtlem15 30550 . . . . . 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 30543 . . . . . . . 8  |-  ( z  .~  w  <->  E. v  e.  A  ( z  e.  v  /\  w  e.  v ) )
81prtlem13 30543 . . . . . . . 8  |-  ( w  .~  p  <->  E. q  e.  A  ( w  e.  q  /\  p  e.  q ) )
97, 8anbi12i 697 . . . . . . 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 3034 . . . . . . 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 252 . . . . . 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 30543 . . . . . 6  |-  ( z  .~  p  <->  E. r  e.  A  ( z  e.  r  /\  p  e.  r ) )
136, 11, 123imtr4g 270 . . . . 5  |-  ( Prt 
A  ->  ( (
z  .~  w  /\  w  .~  p )  -> 
z  .~  p )
)
14 pm3.22 449 . . . . . . 7  |-  ( ( z  e.  v  /\  w  e.  v )  ->  ( w  e.  v  /\  z  e.  v ) )
1514reximi 2935 . . . . . 6  |-  ( E. v  e.  A  ( z  e.  v  /\  w  e.  v )  ->  E. v  e.  A  ( w  e.  v  /\  z  e.  v
) )
161prtlem13 30543 . . . . . 6  |-  ( w  .~  z  <->  E. v  e.  A  ( w  e.  v  /\  z  e.  v ) )
1715, 7, 163imtr4i 266 . . . . 5  |-  ( z  .~  w  ->  w  .~  z )
1813, 17jctil 537 . . . 4  |-  ( Prt 
A  ->  ( (
z  .~  w  ->  w  .~  z )  /\  ( ( z  .~  w  /\  w  .~  p
)  ->  z  .~  p ) ) )
1918alrimivv 1696 . . 3  |-  ( Prt 
A  ->  A. w A. p ( ( z  .~  w  ->  w  .~  z )  /\  (
( z  .~  w  /\  w  .~  p
)  ->  z  .~  p ) ) )
2019alrimiv 1695 . 2  |-  ( Prt 
A  ->  A. z A. w A. p ( ( z  .~  w  ->  w  .~  z )  /\  ( ( z  .~  w  /\  w  .~  p )  ->  z  .~  p ) ) )
21 dfer2 7324 . 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 1180 1  |-  ( Prt 
A  ->  .~  Er  U. A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369   A.wal 1377    = wceq 1379   E.wrex 2818   U.cuni 4251   class class class wbr 4453   {copab 4510   dom cdm 5005   Rel wrel 5010    Er wer 7320   Prt wprt 30546
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pr 4692
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-br 4454  df-opab 4512  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-er 7323  df-prt 30547
This theorem is referenced by:  prtex  30555
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