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Theorem erth 6908
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 6878 . . . . . . . 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 6879 . . . . . . 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 6879 . . . . . 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 2919 . . . . 5  |-  x  e. 
_V
13 erth.2 . . . . . 6  |-  ( ph  ->  A  e.  X )
1413adantr 452 . . . . 5  |-  ( (
ph  /\  A R B )  ->  A  e.  X )
15 elecg 6902 . . . . 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 6873 . . . . . . 7  |-  ( R  Er  X  ->  Rel  R )
182, 17syl 16 . . . . . 6  |-  ( ph  ->  Rel  R )
19 brrelex2 4876 . . . . . 6  |-  ( ( Rel  R  /\  A R B )  ->  B  e.  _V )
2018, 19sylan 458 . . . . 5  |-  ( (
ph  /\  A R B )  ->  B  e.  _V )
21 elecg 6902 . . . . 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 2402 . 2  |-  ( (
ph  /\  A R B )  ->  [ A ] R  =  [ B ] R )
252adantr 452 . . 3  |-  ( (
ph  /\  [ A ] R  =  [ B ] R )  ->  R  Er  X )
262, 13erref 6884 . . . . . . 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 6902 . . . . . . 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 2480 . . . 4  |-  ( (
ph  /\  [ A ] R  =  [ B ] R )  ->  A  e.  [ B ] R )
3425, 32ereldm 6907 . . . . . 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 6902 . . . . 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 6876 . 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 1721   _Vcvv 2916   class class class wbr 4172   Rel wrel 4842    Er wer 6861   [cec 6862
This theorem is referenced by:  erth2  6909  erthi  6910  qliftfun  6948  eroveu  6958  eceqoveq  6968  th3qlem1  6969  enreceq  8900  ercpbllem  13728  orbsta  15045  sylow2blem3  15211  frgpnabllem2  15440  zndvds  16785  divstgpopn  18102  divstgphaus  18105  pi1xfrf  19031  pi1cof  19037  pstmxmet  24245  sconpi1  24879  topfneec2  26262
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 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pr 4363
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 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-sbc 3122  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-br 4173  df-opab 4227  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-er 6864  df-ec 6866
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