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

Theorem iserd 7337
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 2468 . . 3  |-  ( ph  ->  dom  R  =  dom  R )
3 iserd.2 . . . . . . . 8  |-  ( (
ph  /\  x R
y )  ->  y R x )
43ex 434 . . . . . . 7  |-  ( ph  ->  ( x R y  ->  y R x ) )
5 iserd.3 . . . . . . . 8  |-  ( (
ph  /\  ( x R y  /\  y R z ) )  ->  x R z )
65ex 434 . . . . . . 7  |-  ( ph  ->  ( ( x R y  /\  y R z )  ->  x R z ) )
74, 6jca 532 . . . . . 6  |-  ( ph  ->  ( ( x R y  ->  y R x )  /\  (
( x R y  /\  y R z )  ->  x R
z ) ) )
87alrimiv 1695 . . . . 5  |-  ( ph  ->  A. z ( ( x R y  -> 
y R x )  /\  ( ( x R y  /\  y R z )  ->  x R z ) ) )
98alrimiv 1695 . . . 4  |-  ( ph  ->  A. y A. z
( ( x R y  ->  y R x )  /\  (
( x R y  /\  y R z )  ->  x R
z ) ) )
109alrimiv 1695 . . 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 7312 . . 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 1180 . 2  |-  ( ph  ->  R  Er  dom  R
)
1312adantr 465 . . . . . . . 8  |-  ( (
ph  /\  x  e.  dom  R )  ->  R  Er  dom  R )
14 simpr 461 . . . . . . . 8  |-  ( (
ph  /\  x  e.  dom  R )  ->  x  e.  dom  R )
1513, 14erref 7331 . . . . . . 7  |-  ( (
ph  /\  x  e.  dom  R )  ->  x R x )
1615ex 434 . . . . . 6  |-  ( ph  ->  ( x  e.  dom  R  ->  x R x ) )
17 vex 3116 . . . . . . 7  |-  x  e. 
_V
1817, 17breldm 5207 . . . . . 6  |-  ( x R x  ->  x  e.  dom  R )
1916, 18impbid1 203 . . . . 5  |-  ( ph  ->  ( x  e.  dom  R  <-> 
x R x ) )
20 iserd.4 . . . . 5  |-  ( ph  ->  ( x  e.  A  <->  x R x ) )
2119, 20bitr4d 256 . . . 4  |-  ( ph  ->  ( x  e.  dom  R  <-> 
x  e.  A ) )
2221eqrdv 2464 . . 3  |-  ( ph  ->  dom  R  =  A )
23 ereq2 7319 . . 3  |-  ( dom 
R  =  A  -> 
( R  Er  dom  R  <-> 
R  Er  A ) )
2422, 23syl 16 . 2  |-  ( ph  ->  ( R  Er  dom  R  <-> 
R  Er  A ) )
2512, 24mpbid 210 1  |-  ( ph  ->  R  Er  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369   A.wal 1377    = wceq 1379    e. wcel 1767   class class class wbr 4447   dom cdm 4999   Rel wrel 5004    Er wer 7308
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-br 4448  df-opab 4506  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-er 7311
This theorem is referenced by:  swoer  7339  eqer  7344  0er  7346  iiner  7383  erinxp  7385  ecopover  7415  ener  7562  eqger  16056  gicer  16129  gaorber  16151  efgrelexlemb  16574  efgcpbllemb  16579  hmpher  20048  xmeter  20699  phtpcer  21258  vitalilem1  21780  ercgrg  23664  erclwwlk  24520  erclwwlkn  24532  metider  27537
  Copyright terms: Public domain W3C validator