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Theorem erth 6527
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 108 . . . . . . 7  |-  ( (
ph  /\  A R B )  ->  ph )
2 erth.1 . . . . . . . . 9  |-  ( ph  ->  R  Er  X )
32ersymb 6497 . . . . . . . 8  |-  ( ph  ->  ( A R B  <-> 
B R A ) )
43biimpa 294 . . . . . . 7  |-  ( (
ph  /\  A R B )  ->  B R A )
51, 4jca 304 . . . . . 6  |-  ( (
ph  /\  A R B )  ->  ( ph  /\  B R A ) )
62ertr 6498 . . . . . . 7  |-  ( ph  ->  ( ( B R A  /\  A R x )  ->  B R x ) )
76impl 378 . . . . . 6  |-  ( ( ( ph  /\  B R A )  /\  A R x )  ->  B R x )
85, 7sylan 281 . . . . 5  |-  ( ( ( ph  /\  A R B )  /\  A R x )  ->  B R x )
92ertr 6498 . . . . . 6  |-  ( ph  ->  ( ( A R B  /\  B R x )  ->  A R x ) )
109impl 378 . . . . 5  |-  ( ( ( ph  /\  A R B )  /\  B R x )  ->  A R x )
118, 10impbida 586 . . . 4  |-  ( (
ph  /\  A R B )  ->  ( A R x  <->  B R x ) )
12 vex 2715 . . . . 5  |-  x  e. 
_V
13 erth.2 . . . . . 6  |-  ( ph  ->  A  e.  X )
1413adantr 274 . . . . 5  |-  ( (
ph  /\  A R B )  ->  A  e.  X )
15 elecg 6521 . . . . 5  |-  ( ( x  e.  _V  /\  A  e.  X )  ->  ( x  e.  [ A ] R  <->  A R x ) )
1612, 14, 15sylancr 411 . . . 4  |-  ( (
ph  /\  A R B )  ->  (
x  e.  [ A ] R  <->  A R x ) )
17 errel 6492 . . . . . . 7  |-  ( R  Er  X  ->  Rel  R )
182, 17syl 14 . . . . . 6  |-  ( ph  ->  Rel  R )
19 brrelex2 4630 . . . . . 6  |-  ( ( Rel  R  /\  A R B )  ->  B  e.  _V )
2018, 19sylan 281 . . . . 5  |-  ( (
ph  /\  A R B )  ->  B  e.  _V )
21 elecg 6521 . . . . 5  |-  ( ( x  e.  _V  /\  B  e.  _V )  ->  ( x  e.  [ B ] R  <->  B R x ) )
2212, 20, 21sylancr 411 . . . 4  |-  ( (
ph  /\  A R B )  ->  (
x  e.  [ B ] R  <->  B R x ) )
2311, 16, 223bitr4d 219 . . 3  |-  ( (
ph  /\  A R B )  ->  (
x  e.  [ A ] R  <->  x  e.  [ B ] R ) )
2423eqrdv 2155 . 2  |-  ( (
ph  /\  A R B )  ->  [ A ] R  =  [ B ] R )
252adantr 274 . . 3  |-  ( (
ph  /\  [ A ] R  =  [ B ] R )  ->  R  Er  X )
262, 13erref 6503 . . . . . . 7  |-  ( ph  ->  A R A )
2726adantr 274 . . . . . 6  |-  ( (
ph  /\  [ A ] R  =  [ B ] R )  ->  A R A )
2813adantr 274 . . . . . . 7  |-  ( (
ph  /\  [ A ] R  =  [ B ] R )  ->  A  e.  X )
29 elecg 6521 . . . . . . 7  |-  ( ( A  e.  X  /\  A  e.  X )  ->  ( A  e.  [ A ] R  <->  A R A ) )
3028, 28, 29syl2anc 409 . . . . . 6  |-  ( (
ph  /\  [ A ] R  =  [ B ] R )  -> 
( A  e.  [ A ] R  <->  A R A ) )
3127, 30mpbird 166 . . . . 5  |-  ( (
ph  /\  [ A ] R  =  [ B ] R )  ->  A  e.  [ A ] R )
32 simpr 109 . . . . 5  |-  ( (
ph  /\  [ A ] R  =  [ B ] R )  ->  [ A ] R  =  [ B ] R
)
3331, 32eleqtrd 2236 . . . 4  |-  ( (
ph  /\  [ A ] R  =  [ B ] R )  ->  A  e.  [ B ] R )
3425, 32ereldm 6526 . . . . . 6  |-  ( (
ph  /\  [ A ] R  =  [ B ] R )  -> 
( A  e.  X  <->  B  e.  X ) )
3528, 34mpbid 146 . . . . 5  |-  ( (
ph  /\  [ A ] R  =  [ B ] R )  ->  B  e.  X )
36 elecg 6521 . . . . 5  |-  ( ( A  e.  X  /\  B  e.  X )  ->  ( A  e.  [ B ] R  <->  B R A ) )
3728, 35, 36syl2anc 409 . . . 4  |-  ( (
ph  /\  [ A ] R  =  [ B ] R )  -> 
( A  e.  [ B ] R  <->  B R A ) )
3833, 37mpbid 146 . . 3  |-  ( (
ph  /\  [ A ] R  =  [ B ] R )  ->  B R A )
3925, 38ersym 6495 . 2  |-  ( (
ph  /\  [ A ] R  =  [ B ] R )  ->  A R B )
4024, 39impbida 586 1  |-  ( ph  ->  ( A R B  <->  [ A ] R  =  [ B ] R
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    = wceq 1335    e. wcel 2128   _Vcvv 2712   class class class wbr 3967   Rel wrel 4594    Er wer 6480   [cec 6481
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-14 2131  ax-ext 2139  ax-sep 4085  ax-pow 4138  ax-pr 4172
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-v 2714  df-sbc 2938  df-un 3106  df-in 3108  df-ss 3115  df-pw 3546  df-sn 3567  df-pr 3568  df-op 3570  df-br 3968  df-opab 4029  df-xp 4595  df-rel 4596  df-cnv 4597  df-co 4598  df-dm 4599  df-rn 4600  df-res 4601  df-ima 4602  df-er 6483  df-ec 6485
This theorem is referenced by:  erth2  6528  erthi  6529  qliftfun  6565  eroveu  6574  th3qlem1  6585  enqeceq  7282  enq0eceq  7360  nnnq0lem1  7369  enreceq  7659  prsrlem1  7665
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