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Theorem iserd 6068
Description: A reflexive, symmetric, transitive relation is an equivalence relation on its domain. (Contributed by Mario Carneiro, 9-Jul-2014.) (Revised by Mario Carneiro, 12-Aug-2015.)
Hypotheses
Ref Expression
iserd.1  Rel  R
iserd.2  R  R
iserd.3  R  R  R
iserd.4  R
Assertion
Ref Expression
iserd  R  Er
Distinct variable groups:   ,,, R   ,   ,,,
Allowed substitution hints:   (,)

Proof of Theorem iserd
StepHypRef Expression
1 iserd.1 . . 3  Rel  R
2 eqidd 2038 . . 3  dom  R  dom  R
3 iserd.2 . . . . . . . 8  R  R
43ex 108 . . . . . . 7  R  R
5 iserd.3 . . . . . . . 8  R  R  R
65ex 108 . . . . . . 7  R  R  R
74, 6jca 290 . . . . . 6  R  R  R  R  R
87alrimiv 1751 . . . . 5  R  R  R  R  R
98alrimiv 1751 . . . 4  R  R  R  R  R
109alrimiv 1751 . . 3  R  R  R  R  R
11 dfer2 6043 . . 3  R  Er  dom  R  Rel 
R  dom  R 
dom  R  R  R  R  R  R
121, 2, 10, 11syl3anbrc 1087 . 2  R  Er  dom  R
1312adantr 261 . . . . . . . 8  dom  R  R  Er  dom  R
14 simpr 103 . . . . . . . 8  dom  R 
dom  R
1513, 14erref 6062 . . . . . . 7  dom  R  R
1615ex 108 . . . . . 6  dom  R  R
17 vex 2554 . . . . . . 7 
_V
1817, 17breldm 4482 . . . . . 6  R  dom  R
1916, 18impbid1 130 . . . . 5  dom  R  R
20 iserd.4 . . . . 5  R
2119, 20bitr4d 180 . . . 4  dom  R
2221eqrdv 2035 . . 3  dom  R
23 ereq2 6050 . . 3  dom 
R  R  Er  dom  R  R  Er
2422, 23syl 14 . 2  R  Er  dom  R  R  Er
2512, 24mpbid 135 1  R  Er
Colors of variables: wff set class
Syntax hints:   wi 4   wa 97   wb 98  wal 1240   wceq 1242   wcel 1390   class class class wbr 3755   dom cdm 4288   Rel wrel 4293    Er wer 6039
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-er 6042
This theorem is referenced by:  swoer  6070  eqer  6074  0er  6076  iinerm  6114  erinxp  6116  ecopover  6140  ecopoverg  6143  ener  6195  enq0er  6417
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