MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  iserd Unicode version

Theorem iserd 6686
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 2284 . . 3  |-  ( ph  ->  dom  R  =  dom  R )
3 iserd.2 . . . . . . . 8  |-  ( (
ph  /\  x R
y )  ->  y R x )
43ex 423 . . . . . . 7  |-  ( ph  ->  ( x R y  ->  y R x ) )
5 iserd.3 . . . . . . . 8  |-  ( (
ph  /\  ( x R y  /\  y R z ) )  ->  x R z )
65ex 423 . . . . . . 7  |-  ( ph  ->  ( ( x R y  /\  y R z )  ->  x R z ) )
74, 6jca 518 . . . . . 6  |-  ( ph  ->  ( ( x R y  ->  y R x )  /\  (
( x R y  /\  y R z )  ->  x R
z ) ) )
87alrimiv 1617 . . . . 5  |-  ( ph  ->  A. z ( ( x R y  -> 
y R x )  /\  ( ( x R y  /\  y R z )  ->  x R z ) ) )
98alrimiv 1617 . . . 4  |-  ( ph  ->  A. y A. z
( ( x R y  ->  y R x )  /\  (
( x R y  /\  y R z )  ->  x R
z ) ) )
109alrimiv 1617 . . 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 6661 . . 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 1136 . 2  |-  ( ph  ->  R  Er  dom  R
)
1312adantr 451 . . . . . . . 8  |-  ( (
ph  /\  x  e.  dom  R )  ->  R  Er  dom  R )
14 simpr 447 . . . . . . . 8  |-  ( (
ph  /\  x  e.  dom  R )  ->  x  e.  dom  R )
1513, 14erref 6680 . . . . . . 7  |-  ( (
ph  /\  x  e.  dom  R )  ->  x R x )
1615ex 423 . . . . . 6  |-  ( ph  ->  ( x  e.  dom  R  ->  x R x ) )
17 vex 2791 . . . . . . 7  |-  x  e. 
_V
1817, 17breldm 4883 . . . . . 6  |-  ( x R x  ->  x  e.  dom  R )
1916, 18impbid1 194 . . . . 5  |-  ( ph  ->  ( x  e.  dom  R  <-> 
x R x ) )
20 iserd.4 . . . . 5  |-  ( ph  ->  ( x  e.  A  <->  x R x ) )
2119, 20bitr4d 247 . . . 4  |-  ( ph  ->  ( x  e.  dom  R  <-> 
x  e.  A ) )
2221eqrdv 2281 . . 3  |-  ( ph  ->  dom  R  =  A )
23 ereq2 6668 . . 3  |-  ( dom 
R  =  A  -> 
( R  Er  dom  R  <-> 
R  Er  A ) )
2422, 23syl 15 . 2  |-  ( ph  ->  ( R  Er  dom  R  <-> 
R  Er  A ) )
2512, 24mpbid 201 1  |-  ( ph  ->  R  Er  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358   A.wal 1527    = wceq 1623    e. wcel 1684   class class class wbr 4023   dom cdm 4689   Rel wrel 4694    Er wer 6657
This theorem is referenced by:  swoer  6688  eqer  6693  0er  6695  iiner  6731  erinxp  6733  ecopover  6762  ener  6908  eqger  14667  gicer  14740  gaorber  14762  efgrelexlemb  15059  efgcpbllemb  15064  hmpher  17475  xmeter  17979  phtpcer  18493  vitalilem1  18963
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-opab 4078  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-er 6660
  Copyright terms: Public domain W3C validator