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Theorem iserd 7397
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 2430 . . 3  |-  ( ph  ->  dom  R  =  dom  R )
3 iserd.2 . . . . . . . 8  |-  ( (
ph  /\  x R
y )  ->  y R x )
43ex 435 . . . . . . 7  |-  ( ph  ->  ( x R y  ->  y R x ) )
5 iserd.3 . . . . . . . 8  |-  ( (
ph  /\  ( x R y  /\  y R z ) )  ->  x R z )
65ex 435 . . . . . . 7  |-  ( ph  ->  ( ( x R y  /\  y R z )  ->  x R z ) )
74, 6jca 534 . . . . . 6  |-  ( ph  ->  ( ( x R y  ->  y R x )  /\  (
( x R y  /\  y R z )  ->  x R
z ) ) )
87alrimiv 1766 . . . . 5  |-  ( ph  ->  A. z ( ( x R y  -> 
y R x )  /\  ( ( x R y  /\  y R z )  ->  x R z ) ) )
98alrimiv 1766 . . . 4  |-  ( ph  ->  A. y A. z
( ( x R y  ->  y R x )  /\  (
( x R y  /\  y R z )  ->  x R
z ) ) )
109alrimiv 1766 . . 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 7372 . . 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 1189 . 2  |-  ( ph  ->  R  Er  dom  R
)
1312adantr 466 . . . . . . . 8  |-  ( (
ph  /\  x  e.  dom  R )  ->  R  Er  dom  R )
14 simpr 462 . . . . . . . 8  |-  ( (
ph  /\  x  e.  dom  R )  ->  x  e.  dom  R )
1513, 14erref 7391 . . . . . . 7  |-  ( (
ph  /\  x  e.  dom  R )  ->  x R x )
1615ex 435 . . . . . 6  |-  ( ph  ->  ( x  e.  dom  R  ->  x R x ) )
17 vex 3090 . . . . . . 7  |-  x  e. 
_V
1817, 17breldm 5059 . . . . . 6  |-  ( x R x  ->  x  e.  dom  R )
1916, 18impbid1 206 . . . . 5  |-  ( ph  ->  ( x  e.  dom  R  <-> 
x R x ) )
20 iserd.4 . . . . 5  |-  ( ph  ->  ( x  e.  A  <->  x R x ) )
2119, 20bitr4d 259 . . . 4  |-  ( ph  ->  ( x  e.  dom  R  <-> 
x  e.  A ) )
2221eqrdv 2426 . . 3  |-  ( ph  ->  dom  R  =  A )
23 ereq2 7379 . . 3  |-  ( dom 
R  =  A  -> 
( R  Er  dom  R  <-> 
R  Er  A ) )
2422, 23syl 17 . 2  |-  ( ph  ->  ( R  Er  dom  R  <-> 
R  Er  A ) )
2512, 24mpbid 213 1  |-  ( ph  ->  R  Er  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370   A.wal 1435    = wceq 1437    e. wcel 1870   class class class wbr 4426   dom cdm 4854   Rel wrel 4859    Er wer 7368
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pr 4661
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-rab 2791  df-v 3089  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-sn 4003  df-pr 4005  df-op 4009  df-br 4427  df-opab 4485  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-er 7371
This theorem is referenced by:  swoer  7399  eqer  7404  0er  7406  iiner  7443  erinxp  7445  ecopover  7475  ener  7623  cicer  15662  eqger  16818  gicer  16891  gaorber  16913  efgrelexlemb  17335  efgcpbllemb  17340  hmpher  20730  xmeter  21379  phtpcer  21919  vitalilem1  22443  ercgrg  24424  erclwwlk  25389  erclwwlkn  25401  metider  28536
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