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Theorem erth 6887
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
) )

Proof of Theorem erth
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 simpl 444 . . . . . . 7  |-  ( (
ph  /\  A R B )  ->  ph )
2 erth.1 . . . . . . . . 9  |-  ( ph  ->  R  Er  X )
32ersymb 6857 . . . . . . . 8  |-  ( ph  ->  ( A R B  <-> 
B R A ) )
43biimpa 471 . . . . . . 7  |-  ( (
ph  /\  A R B )  ->  B R A )
51, 4jca 519 . . . . . 6  |-  ( (
ph  /\  A R B )  ->  ( ph  /\  B R A ) )
62ertr 6858 . . . . . . 7  |-  ( ph  ->  ( ( B R A  /\  A R x )  ->  B R x ) )
76impl 604 . . . . . 6  |-  ( ( ( ph  /\  B R A )  /\  A R x )  ->  B R x )
85, 7sylan 458 . . . . 5  |-  ( ( ( ph  /\  A R B )  /\  A R x )  ->  B R x )
92ertr 6858 . . . . . 6  |-  ( ph  ->  ( ( A R B  /\  B R x )  ->  A R x ) )
109impl 604 . . . . 5  |-  ( ( ( ph  /\  A R B )  /\  B R x )  ->  A R x )
118, 10impbida 806 . . . 4  |-  ( (
ph  /\  A R B )  ->  ( A R x  <->  B R x ) )
12 vex 2904 . . . . 5  |-  x  e. 
_V
13 erth.2 . . . . . 6  |-  ( ph  ->  A  e.  X )
1413adantr 452 . . . . 5  |-  ( (
ph  /\  A R B )  ->  A  e.  X )
15 elecg 6881 . . . . 5  |-  ( ( x  e.  _V  /\  A  e.  X )  ->  ( x  e.  [ A ] R  <->  A R x ) )
1612, 14, 15sylancr 645 . . . 4  |-  ( (
ph  /\  A R B )  ->  (
x  e.  [ A ] R  <->  A R x ) )
17 errel 6852 . . . . . . 7  |-  ( R  Er  X  ->  Rel  R )
182, 17syl 16 . . . . . 6  |-  ( ph  ->  Rel  R )
19 brrelex2 4859 . . . . . 6  |-  ( ( Rel  R  /\  A R B )  ->  B  e.  _V )
2018, 19sylan 458 . . . . 5  |-  ( (
ph  /\  A R B )  ->  B  e.  _V )
21 elecg 6881 . . . . 5  |-  ( ( x  e.  _V  /\  B  e.  _V )  ->  ( x  e.  [ B ] R  <->  B R x ) )
2212, 20, 21sylancr 645 . . . 4  |-  ( (
ph  /\  A R B )  ->  (
x  e.  [ B ] R  <->  B R x ) )
2311, 16, 223bitr4d 277 . . 3  |-  ( (
ph  /\  A R B )  ->  (
x  e.  [ A ] R  <->  x  e.  [ B ] R ) )
2423eqrdv 2387 . 2  |-  ( (
ph  /\  A R B )  ->  [ A ] R  =  [ B ] R )
252adantr 452 . . 3  |-  ( (
ph  /\  [ A ] R  =  [ B ] R )  ->  R  Er  X )
262, 13erref 6863 . . . . . . 7  |-  ( ph  ->  A R A )
2726adantr 452 . . . . . 6  |-  ( (
ph  /\  [ A ] R  =  [ B ] R )  ->  A R A )
2813adantr 452 . . . . . . 7  |-  ( (
ph  /\  [ A ] R  =  [ B ] R )  ->  A  e.  X )
29 elecg 6881 . . . . . . 7  |-  ( ( A  e.  X  /\  A  e.  X )  ->  ( A  e.  [ A ] R  <->  A R A ) )
3028, 28, 29syl2anc 643 . . . . . 6  |-  ( (
ph  /\  [ A ] R  =  [ B ] R )  -> 
( A  e.  [ A ] R  <->  A R A ) )
3127, 30mpbird 224 . . . . 5  |-  ( (
ph  /\  [ A ] R  =  [ B ] R )  ->  A  e.  [ A ] R )
32 simpr 448 . . . . 5  |-  ( (
ph  /\  [ A ] R  =  [ B ] R )  ->  [ A ] R  =  [ B ] R
)
3331, 32eleqtrd 2465 . . . 4  |-  ( (
ph  /\  [ A ] R  =  [ B ] R )  ->  A  e.  [ B ] R )
3425, 32ereldm 6886 . . . . . 6  |-  ( (
ph  /\  [ A ] R  =  [ B ] R )  -> 
( A  e.  X  <->  B  e.  X ) )
3528, 34mpbid 202 . . . . 5  |-  ( (
ph  /\  [ A ] R  =  [ B ] R )  ->  B  e.  X )
36 elecg 6881 . . . . 5  |-  ( ( A  e.  X  /\  B  e.  X )  ->  ( A  e.  [ B ] R  <->  B R A ) )
3728, 35, 36syl2anc 643 . . . 4  |-  ( (
ph  /\  [ A ] R  =  [ B ] R )  -> 
( A  e.  [ B ] R  <->  B R A ) )
3833, 37mpbid 202 . . 3  |-  ( (
ph  /\  [ A ] R  =  [ B ] R )  ->  B R A )
3925, 38ersym 6855 . 2  |-  ( (
ph  /\  [ A ] R  =  [ B ] R )  ->  A R B )
4024, 39impbida 806 1  |-  ( ph  ->  ( A R B  <->  [ A ] R  =  [ B ] R
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1717   _Vcvv 2901   class class class wbr 4155   Rel wrel 4825    Er wer 6840   [cec 6841
This theorem is referenced by:  erth2  6888  erthi  6889  qliftfun  6927  eroveu  6937  eceqoveq  6947  th3qlem1  6948  enreceq  8879  ercpbllem  13702  orbsta  15019  sylow2blem3  15185  frgpnabllem2  15414  zndvds  16755  divstgpopn  18072  divstgphaus  18075  pi1xfrf  18951  pi1cof  18957  sconpi1  24707  topfneec2  26065
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-sep 4273  ax-nul 4281  ax-pr 4346
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-ral 2656  df-rex 2657  df-rab 2660  df-v 2903  df-sbc 3107  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-nul 3574  df-if 3685  df-sn 3765  df-pr 3766  df-op 3768  df-br 4156  df-opab 4210  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-er 6843  df-ec 6845
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