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Theorem eqer 6709
Description: Equivalence relation involving equality of dependent classes  A ( x ) and  B ( y ). (Contributed by NM, 17-Mar-2008.) (Revised by Mario Carneiro, 12-Aug-2015.)
Hypotheses
Ref Expression
eqer.1  |-  ( x  =  y  ->  A  =  B )
eqer.2  |-  R  =  { <. x ,  y
>.  |  A  =  B }
Assertion
Ref Expression
eqer  |-  R  Er  _V
Distinct variable groups:    x, y    y, A    x, B
Allowed substitution hints:    A( x)    B( y)    R( x, y)

Proof of Theorem eqer
Dummy variables  w  z  v are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqer.2 . . . . 5  |-  R  =  { <. x ,  y
>.  |  A  =  B }
21relopabi 4827 . . . 4  |-  Rel  R
32a1i 10 . . 3  |-  (  T. 
->  Rel  R )
4 id 19 . . . . . 6  |-  ( [_ z  /  x ]_ A  =  [_ w  /  x ]_ A  ->  [_ z  /  x ]_ A  = 
[_ w  /  x ]_ A )
54eqcomd 2301 . . . . 5  |-  ( [_ z  /  x ]_ A  =  [_ w  /  x ]_ A  ->  [_ w  /  x ]_ A  = 
[_ z  /  x ]_ A )
6 eqer.1 . . . . . 6  |-  ( x  =  y  ->  A  =  B )
76, 1eqerlem 6708 . . . . 5  |-  ( z R w  <->  [_ z  /  x ]_ A  =  [_ w  /  x ]_ A
)
86, 1eqerlem 6708 . . . . 5  |-  ( w R z  <->  [_ w  /  x ]_ A  =  [_ z  /  x ]_ A
)
95, 7, 83imtr4i 257 . . . 4  |-  ( z R w  ->  w R z )
109adantl 452 . . 3  |-  ( (  T.  /\  z R w )  ->  w R z )
11 eqtr 2313 . . . . 5  |-  ( (
[_ z  /  x ]_ A  =  [_ w  /  x ]_ A  /\  [_ w  /  x ]_ A  =  [_ v  /  x ]_ A )  ->  [_ z  /  x ]_ A  =  [_ v  /  x ]_ A )
126, 1eqerlem 6708 . . . . . 6  |-  ( w R v  <->  [_ w  /  x ]_ A  =  [_ v  /  x ]_ A
)
137, 12anbi12i 678 . . . . 5  |-  ( ( z R w  /\  w R v )  <->  ( [_ z  /  x ]_ A  =  [_ w  /  x ]_ A  /\  [_ w  /  x ]_ A  = 
[_ v  /  x ]_ A ) )
146, 1eqerlem 6708 . . . . 5  |-  ( z R v  <->  [_ z  /  x ]_ A  =  [_ v  /  x ]_ A
)
1511, 13, 143imtr4i 257 . . . 4  |-  ( ( z R w  /\  w R v )  -> 
z R v )
1615adantl 452 . . 3  |-  ( (  T.  /\  ( z R w  /\  w R v ) )  ->  z R v )
17 vex 2804 . . . . 5  |-  z  e. 
_V
18 eqid 2296 . . . . . 6  |-  [_ z  /  x ]_ A  = 
[_ z  /  x ]_ A
196, 1eqerlem 6708 . . . . . 6  |-  ( z R z  <->  [_ z  /  x ]_ A  =  [_ z  /  x ]_ A
)
2018, 19mpbir 200 . . . . 5  |-  z R z
2117, 202th 230 . . . 4  |-  ( z  e.  _V  <->  z R
z )
2221a1i 10 . . 3  |-  (  T. 
->  ( z  e.  _V  <->  z R z ) )
233, 10, 16, 22iserd 6702 . 2  |-  (  T. 
->  R  Er  _V )
2423trud 1314 1  |-  R  Er  _V
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    T. wtru 1307    = wceq 1632    e. wcel 1696   _Vcvv 2801   [_csb 3094   class class class wbr 4039   {copab 4092   Rel wrel 4710    Er wer 6673
This theorem is referenced by:  ider  6710  frgpuplem  15097  fneer  26391
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-br 4040  df-opab 4094  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-er 6676
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