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Theorem iserd 6421
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 2116 . . 3  |-  ( ph  ->  dom  R  =  dom  R )
3 iserd.2 . . . . . . . 8  |-  ( (
ph  /\  x R
y )  ->  y R x )
43ex 114 . . . . . . 7  |-  ( ph  ->  ( x R y  ->  y R x ) )
5 iserd.3 . . . . . . . 8  |-  ( (
ph  /\  ( x R y  /\  y R z ) )  ->  x R z )
65ex 114 . . . . . . 7  |-  ( ph  ->  ( ( x R y  /\  y R z )  ->  x R z ) )
74, 6jca 302 . . . . . 6  |-  ( ph  ->  ( ( x R y  ->  y R x )  /\  (
( x R y  /\  y R z )  ->  x R
z ) ) )
87alrimiv 1828 . . . . 5  |-  ( ph  ->  A. z ( ( x R y  -> 
y R x )  /\  ( ( x R y  /\  y R z )  ->  x R z ) ) )
98alrimiv 1828 . . . 4  |-  ( ph  ->  A. y A. z
( ( x R y  ->  y R x )  /\  (
( x R y  /\  y R z )  ->  x R
z ) ) )
109alrimiv 1828 . . 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 6396 . . 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 1148 . 2  |-  ( ph  ->  R  Er  dom  R
)
1312adantr 272 . . . . . . . 8  |-  ( (
ph  /\  x  e.  dom  R )  ->  R  Er  dom  R )
14 simpr 109 . . . . . . . 8  |-  ( (
ph  /\  x  e.  dom  R )  ->  x  e.  dom  R )
1513, 14erref 6415 . . . . . . 7  |-  ( (
ph  /\  x  e.  dom  R )  ->  x R x )
1615ex 114 . . . . . 6  |-  ( ph  ->  ( x  e.  dom  R  ->  x R x ) )
17 vex 2661 . . . . . . 7  |-  x  e. 
_V
1817, 17breldm 4711 . . . . . 6  |-  ( x R x  ->  x  e.  dom  R )
1916, 18impbid1 141 . . . . 5  |-  ( ph  ->  ( x  e.  dom  R  <-> 
x R x ) )
20 iserd.4 . . . . 5  |-  ( ph  ->  ( x  e.  A  <->  x R x ) )
2119, 20bitr4d 190 . . . 4  |-  ( ph  ->  ( x  e.  dom  R  <-> 
x  e.  A ) )
2221eqrdv 2113 . . 3  |-  ( ph  ->  dom  R  =  A )
23 ereq2 6403 . . 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 146 1  |-  ( ph  ->  R  Er  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104   A.wal 1312    = wceq 1314    e. wcel 1463   class class class wbr 3897   dom cdm 4507   Rel wrel 4512    Er wer 6392
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 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-pow 4066  ax-pr 4099
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396  df-rex 2397  df-v 2660  df-un 3043  df-in 3045  df-ss 3052  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-br 3898  df-opab 3958  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-er 6395
This theorem is referenced by:  swoer  6423  eqer  6427  0er  6429  iinerm  6467  erinxp  6469  ecopover  6493  ecopoverg  6496  ener  6639  enq0er  7207  xmeter  12500
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