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Theorem iserd 6163
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  |-  ( ph  ->  Rel  R )
iserd.2  |-  ( (
ph  /\  x R
y )  ->  y R x )
iserd.3  |-  ( (
ph  /\  ( x R y  /\  y R z ) )  ->  x R z )
iserd.4  |-  ( ph  ->  ( x  e.  A  <->  x R x ) )
Assertion
Ref Expression
iserd  |-  ( ph  ->  R  Er  A )
Distinct variable groups:    x, y, z, R    x, A    ph, x, y, z
Allowed substitution hints:    A( y, z)

Proof of Theorem iserd
StepHypRef Expression
1 iserd.1 . . 3  |-  ( ph  ->  Rel  R )
2 eqidd 2057 . . 3  |-  ( ph  ->  dom  R  =  dom  R )
3 iserd.2 . . . . . . . 8  |-  ( (
ph  /\  x R
y )  ->  y R x )
43ex 112 . . . . . . 7  |-  ( ph  ->  ( x R y  ->  y R x ) )
5 iserd.3 . . . . . . . 8  |-  ( (
ph  /\  ( x R y  /\  y R z ) )  ->  x R z )
65ex 112 . . . . . . 7  |-  ( ph  ->  ( ( x R y  /\  y R z )  ->  x R z ) )
74, 6jca 294 . . . . . 6  |-  ( ph  ->  ( ( x R y  ->  y R x )  /\  (
( x R y  /\  y R z )  ->  x R
z ) ) )
87alrimiv 1770 . . . . 5  |-  ( ph  ->  A. z ( ( x R y  -> 
y R x )  /\  ( ( x R y  /\  y R z )  ->  x R z ) ) )
98alrimiv 1770 . . . 4  |-  ( ph  ->  A. y A. z
( ( x R y  ->  y R x )  /\  (
( x R y  /\  y R z )  ->  x R
z ) ) )
109alrimiv 1770 . . 3  |-  ( ph  ->  A. x A. y A. z ( ( x R y  ->  y R x )  /\  ( ( x R y  /\  y R z )  ->  x R z ) ) )
11 dfer2 6138 . . 3  |-  ( R  Er  dom  R  <->  ( Rel  R  /\  dom  R  =  dom  R  /\  A. x A. y A. z
( ( x R y  ->  y R x )  /\  (
( x R y  /\  y R z )  ->  x R
z ) ) ) )
121, 2, 10, 11syl3anbrc 1099 . 2  |-  ( ph  ->  R  Er  dom  R
)
1312adantr 265 . . . . . . . 8  |-  ( (
ph  /\  x  e.  dom  R )  ->  R  Er  dom  R )
14 simpr 107 . . . . . . . 8  |-  ( (
ph  /\  x  e.  dom  R )  ->  x  e.  dom  R )
1513, 14erref 6157 . . . . . . 7  |-  ( (
ph  /\  x  e.  dom  R )  ->  x R x )
1615ex 112 . . . . . 6  |-  ( ph  ->  ( x  e.  dom  R  ->  x R x ) )
17 vex 2577 . . . . . . 7  |-  x  e. 
_V
1817, 17breldm 4567 . . . . . 6  |-  ( x R x  ->  x  e.  dom  R )
1916, 18impbid1 134 . . . . 5  |-  ( ph  ->  ( x  e.  dom  R  <-> 
x R x ) )
20 iserd.4 . . . . 5  |-  ( ph  ->  ( x  e.  A  <->  x R x ) )
2119, 20bitr4d 184 . . . 4  |-  ( ph  ->  ( x  e.  dom  R  <-> 
x  e.  A ) )
2221eqrdv 2054 . . 3  |-  ( ph  ->  dom  R  =  A )
23 ereq2 6145 . . 3  |-  ( dom 
R  =  A  -> 
( R  Er  dom  R  <-> 
R  Er  A ) )
2422, 23syl 14 . 2  |-  ( ph  ->  ( R  Er  dom  R  <-> 
R  Er  A ) )
2512, 24mpbid 139 1  |-  ( ph  ->  R  Er  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 101    <-> wb 102   A.wal 1257    = wceq 1259    e. wcel 1409   class class class wbr 3792   dom cdm 4373   Rel wrel 4378    Er wer 6134
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-pow 3955  ax-pr 3972
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-un 2950  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-br 3793  df-opab 3847  df-xp 4379  df-rel 4380  df-cnv 4381  df-co 4382  df-dm 4383  df-er 6137
This theorem is referenced by:  swoer  6165  eqer  6169  0er  6171  iinerm  6209  erinxp  6211  ecopover  6235  ecopoverg  6238  ener  6290  enq0er  6591
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