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Theorem erth 6578
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 109 . . . . . . 7  |-  ( (
ph  /\  A R B )  ->  ph )
2 erth.1 . . . . . . . . 9  |-  ( ph  ->  R  Er  X )
32ersymb 6548 . . . . . . . 8  |-  ( ph  ->  ( A R B  <-> 
B R A ) )
43biimpa 296 . . . . . . 7  |-  ( (
ph  /\  A R B )  ->  B R A )
51, 4jca 306 . . . . . 6  |-  ( (
ph  /\  A R B )  ->  ( ph  /\  B R A ) )
62ertr 6549 . . . . . . 7  |-  ( ph  ->  ( ( B R A  /\  A R x )  ->  B R x ) )
76impl 380 . . . . . 6  |-  ( ( ( ph  /\  B R A )  /\  A R x )  ->  B R x )
85, 7sylan 283 . . . . 5  |-  ( ( ( ph  /\  A R B )  /\  A R x )  ->  B R x )
92ertr 6549 . . . . . 6  |-  ( ph  ->  ( ( A R B  /\  B R x )  ->  A R x ) )
109impl 380 . . . . 5  |-  ( ( ( ph  /\  A R B )  /\  B R x )  ->  A R x )
118, 10impbida 596 . . . 4  |-  ( (
ph  /\  A R B )  ->  ( A R x  <->  B R x ) )
12 vex 2740 . . . . 5  |-  x  e. 
_V
13 erth.2 . . . . . 6  |-  ( ph  ->  A  e.  X )
1413adantr 276 . . . . 5  |-  ( (
ph  /\  A R B )  ->  A  e.  X )
15 elecg 6572 . . . . 5  |-  ( ( x  e.  _V  /\  A  e.  X )  ->  ( x  e.  [ A ] R  <->  A R x ) )
1612, 14, 15sylancr 414 . . . 4  |-  ( (
ph  /\  A R B )  ->  (
x  e.  [ A ] R  <->  A R x ) )
17 errel 6543 . . . . . . 7  |-  ( R  Er  X  ->  Rel  R )
182, 17syl 14 . . . . . 6  |-  ( ph  ->  Rel  R )
19 brrelex2 4667 . . . . . 6  |-  ( ( Rel  R  /\  A R B )  ->  B  e.  _V )
2018, 19sylan 283 . . . . 5  |-  ( (
ph  /\  A R B )  ->  B  e.  _V )
21 elecg 6572 . . . . 5  |-  ( ( x  e.  _V  /\  B  e.  _V )  ->  ( x  e.  [ B ] R  <->  B R x ) )
2212, 20, 21sylancr 414 . . . 4  |-  ( (
ph  /\  A R B )  ->  (
x  e.  [ B ] R  <->  B R x ) )
2311, 16, 223bitr4d 220 . . 3  |-  ( (
ph  /\  A R B )  ->  (
x  e.  [ A ] R  <->  x  e.  [ B ] R ) )
2423eqrdv 2175 . 2  |-  ( (
ph  /\  A R B )  ->  [ A ] R  =  [ B ] R )
252adantr 276 . . 3  |-  ( (
ph  /\  [ A ] R  =  [ B ] R )  ->  R  Er  X )
262, 13erref 6554 . . . . . . 7  |-  ( ph  ->  A R A )
2726adantr 276 . . . . . 6  |-  ( (
ph  /\  [ A ] R  =  [ B ] R )  ->  A R A )
2813adantr 276 . . . . . . 7  |-  ( (
ph  /\  [ A ] R  =  [ B ] R )  ->  A  e.  X )
29 elecg 6572 . . . . . . 7  |-  ( ( A  e.  X  /\  A  e.  X )  ->  ( A  e.  [ A ] R  <->  A R A ) )
3028, 28, 29syl2anc 411 . . . . . 6  |-  ( (
ph  /\  [ A ] R  =  [ B ] R )  -> 
( A  e.  [ A ] R  <->  A R A ) )
3127, 30mpbird 167 . . . . 5  |-  ( (
ph  /\  [ A ] R  =  [ B ] R )  ->  A  e.  [ A ] R )
32 simpr 110 . . . . 5  |-  ( (
ph  /\  [ A ] R  =  [ B ] R )  ->  [ A ] R  =  [ B ] R
)
3331, 32eleqtrd 2256 . . . 4  |-  ( (
ph  /\  [ A ] R  =  [ B ] R )  ->  A  e.  [ B ] R )
3425, 32ereldm 6577 . . . . . 6  |-  ( (
ph  /\  [ A ] R  =  [ B ] R )  -> 
( A  e.  X  <->  B  e.  X ) )
3528, 34mpbid 147 . . . . 5  |-  ( (
ph  /\  [ A ] R  =  [ B ] R )  ->  B  e.  X )
36 elecg 6572 . . . . 5  |-  ( ( A  e.  X  /\  B  e.  X )  ->  ( A  e.  [ B ] R  <->  B R A ) )
3728, 35, 36syl2anc 411 . . . 4  |-  ( (
ph  /\  [ A ] R  =  [ B ] R )  -> 
( A  e.  [ B ] R  <->  B R A ) )
3833, 37mpbid 147 . . 3  |-  ( (
ph  /\  [ A ] R  =  [ B ] R )  ->  B R A )
3925, 38ersym 6546 . 2  |-  ( (
ph  /\  [ A ] R  =  [ B ] R )  ->  A R B )
4024, 39impbida 596 1  |-  ( ph  ->  ( A R B  <->  [ A ] R  =  [ B ] R
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    <-> wb 105    = wceq 1353    e. wcel 2148   _Vcvv 2737   class class class wbr 4003   Rel wrel 4631    Er wer 6531   [cec 6532
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4121  ax-pow 4174  ax-pr 4209
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-sbc 2963  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-br 4004  df-opab 4065  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-res 4638  df-ima 4639  df-er 6534  df-ec 6536
This theorem is referenced by:  erth2  6579  erthi  6580  qliftfun  6616  eroveu  6625  th3qlem1  6636  enqeceq  7357  enq0eceq  7435  nnnq0lem1  7444  enreceq  7734  prsrlem1  7740  ercpbllemg  12748
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