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Theorem erth 6700
Description: Basic property of equivalence relations. Theorem 73 of [Suppes] p. 82. (Contributed by NM, 23-Jul-1995.) (Revised by Mario Carneiro, 6-Jul-2015.)
Hypotheses
Ref Expression
erth.1  |-  ( ph  ->  R  Er  X )
erth.2  |-  ( ph  ->  A  e.  X )
Assertion
Ref Expression
erth  |-  ( ph  ->  ( A R B  <->  [ A ] R  =  [ B ] R
) )
Dummy variable  x is distinct from all other variables.

Proof of Theorem erth
StepHypRef Expression
1 simpl 445 . . . . . . 7  |-  ( (
ph  /\  A R B )  ->  ph )
2 erth.1 . . . . . . . . 9  |-  ( ph  ->  R  Er  X )
32ersymb 6670 . . . . . . . 8  |-  ( ph  ->  ( A R B  <-> 
B R A ) )
43biimpa 472 . . . . . . 7  |-  ( (
ph  /\  A R B )  ->  B R A )
51, 4jca 520 . . . . . 6  |-  ( (
ph  /\  A R B )  ->  ( ph  /\  B R A ) )
62ertr 6671 . . . . . . 7  |-  ( ph  ->  ( ( B R A  /\  A R x )  ->  B R x ) )
76impl 605 . . . . . 6  |-  ( ( ( ph  /\  B R A )  /\  A R x )  ->  B R x )
85, 7sylan 459 . . . . 5  |-  ( ( ( ph  /\  A R B )  /\  A R x )  ->  B R x )
92ertr 6671 . . . . . 6  |-  ( ph  ->  ( ( A R B  /\  B R x )  ->  A R x ) )
109impl 605 . . . . 5  |-  ( ( ( ph  /\  A R B )  /\  B R x )  ->  A R x )
118, 10impbida 807 . . . 4  |-  ( (
ph  /\  A R B )  ->  ( A R x  <->  B R x ) )
12 vex 2793 . . . . 5  |-  x  e. 
_V
13 erth.2 . . . . . 6  |-  ( ph  ->  A  e.  X )
1413adantr 453 . . . . 5  |-  ( (
ph  /\  A R B )  ->  A  e.  X )
15 elecg 6694 . . . . 5  |-  ( ( x  e.  _V  /\  A  e.  X )  ->  ( x  e.  [ A ] R  <->  A R x ) )
1612, 14, 15sylancr 646 . . . 4  |-  ( (
ph  /\  A R B )  ->  (
x  e.  [ A ] R  <->  A R x ) )
17 errel 6665 . . . . . . 7  |-  ( R  Er  X  ->  Rel  R )
182, 17syl 17 . . . . . 6  |-  ( ph  ->  Rel  R )
19 brrelex2 4728 . . . . . 6  |-  ( ( Rel  R  /\  A R B )  ->  B  e.  _V )
2018, 19sylan 459 . . . . 5  |-  ( (
ph  /\  A R B )  ->  B  e.  _V )
21 elecg 6694 . . . . 5  |-  ( ( x  e.  _V  /\  B  e.  _V )  ->  ( x  e.  [ B ] R  <->  B R x ) )
2212, 20, 21sylancr 646 . . . 4  |-  ( (
ph  /\  A R B )  ->  (
x  e.  [ B ] R  <->  B R x ) )
2311, 16, 223bitr4d 278 . . 3  |-  ( (
ph  /\  A R B )  ->  (
x  e.  [ A ] R  <->  x  e.  [ B ] R ) )
2423eqrdv 2283 . 2  |-  ( (
ph  /\  A R B )  ->  [ A ] R  =  [ B ] R )
252adantr 453 . . 3  |-  ( (
ph  /\  [ A ] R  =  [ B ] R )  ->  R  Er  X )
262, 13erref 6676 . . . . . . 7  |-  ( ph  ->  A R A )
2726adantr 453 . . . . . 6  |-  ( (
ph  /\  [ A ] R  =  [ B ] R )  ->  A R A )
2813adantr 453 . . . . . . 7  |-  ( (
ph  /\  [ A ] R  =  [ B ] R )  ->  A  e.  X )
29 elecg 6694 . . . . . . 7  |-  ( ( A  e.  X  /\  A  e.  X )  ->  ( A  e.  [ A ] R  <->  A R A ) )
3028, 28, 29syl2anc 644 . . . . . 6  |-  ( (
ph  /\  [ A ] R  =  [ B ] R )  -> 
( A  e.  [ A ] R  <->  A R A ) )
3127, 30mpbird 225 . . . . 5  |-  ( (
ph  /\  [ A ] R  =  [ B ] R )  ->  A  e.  [ A ] R )
32 simpr 449 . . . . 5  |-  ( (
ph  /\  [ A ] R  =  [ B ] R )  ->  [ A ] R  =  [ B ] R
)
3331, 32eleqtrd 2361 . . . 4  |-  ( (
ph  /\  [ A ] R  =  [ B ] R )  ->  A  e.  [ B ] R )
3425, 32ereldm 6699 . . . . . 6  |-  ( (
ph  /\  [ A ] R  =  [ B ] R )  -> 
( A  e.  X  <->  B  e.  X ) )
3528, 34mpbid 203 . . . . 5  |-  ( (
ph  /\  [ A ] R  =  [ B ] R )  ->  B  e.  X )
36 elecg 6694 . . . . 5  |-  ( ( A  e.  X  /\  B  e.  X )  ->  ( A  e.  [ B ] R  <->  B R A ) )
3728, 35, 36syl2anc 644 . . . 4  |-  ( (
ph  /\  [ A ] R  =  [ B ] R )  -> 
( A  e.  [ B ] R  <->  B R A ) )
3833, 37mpbid 203 . . 3  |-  ( (
ph  /\  [ A ] R  =  [ B ] R )  ->  B R A )
3925, 38ersym 6668 . 2  |-  ( (
ph  /\  [ A ] R  =  [ B ] R )  ->  A R B )
4024, 39impbida 807 1  |-  ( ph  ->  ( A R B  <->  [ A ] R  =  [ B ] R
) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    <-> wb 178    /\ wa 360    = wceq 1624    e. wcel 1685   _Vcvv 2790   class class class wbr 4025   Rel wrel 4694    Er wer 6653   [cec 6654
This theorem is referenced by:  erth2  6701  erthi  6702  qliftfun  6739  eroveu  6749  eceqoveq  6759  th3qlem1  6760  enreceq  8687  ercpbllem  13445  orbsta  14762  sylow2blem3  14928  frgpnabllem2  15157  zndvds  16498  divstgpopn  17797  divstgphaus  17800  pi1xfrf  18546  pi1cof  18552  sconpi1  23175  topfneec2  25692
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2266  ax-sep 4143  ax-nul 4151  ax-pr 4214
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-ral 2550  df-rex 2551  df-rab 2554  df-v 2792  df-sbc 2994  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-nul 3458  df-if 3568  df-sn 3648  df-pr 3649  df-op 3651  df-br 4026  df-opab 4080  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-er 6656  df-ec 6658
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