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Theorem eqer 7234
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 5063 . . . 4  |-  Rel  R
32a1i 11 . . 3  |-  ( T. 
->  Rel  R )
4 id 22 . . . . . 6  |-  ( [_ z  /  x ]_ A  =  [_ w  /  x ]_ A  ->  [_ z  /  x ]_ A  = 
[_ w  /  x ]_ A )
54eqcomd 2459 . . . . 5  |-  ( [_ z  /  x ]_ A  =  [_ w  /  x ]_ A  ->  [_ w  /  x ]_ A  = 
[_ z  /  x ]_ A )
6 eqer.1 . . . . . 6  |-  ( x  =  y  ->  A  =  B )
76, 1eqerlem 7233 . . . . 5  |-  ( z R w  <->  [_ z  /  x ]_ A  =  [_ w  /  x ]_ A
)
86, 1eqerlem 7233 . . . . 5  |-  ( w R z  <->  [_ w  /  x ]_ A  =  [_ z  /  x ]_ A
)
95, 7, 83imtr4i 266 . . . 4  |-  ( z R w  ->  w R z )
109adantl 466 . . 3  |-  ( ( T.  /\  z R w )  ->  w R z )
11 eqtr 2477 . . . . 5  |-  ( (
[_ z  /  x ]_ A  =  [_ w  /  x ]_ A  /\  [_ w  /  x ]_ A  =  [_ v  /  x ]_ A )  ->  [_ z  /  x ]_ A  =  [_ v  /  x ]_ A )
126, 1eqerlem 7233 . . . . . 6  |-  ( w R v  <->  [_ w  /  x ]_ A  =  [_ v  /  x ]_ A
)
137, 12anbi12i 697 . . . . 5  |-  ( ( z R w  /\  w R v )  <->  ( [_ z  /  x ]_ A  =  [_ w  /  x ]_ A  /\  [_ w  /  x ]_ A  = 
[_ v  /  x ]_ A ) )
146, 1eqerlem 7233 . . . . 5  |-  ( z R v  <->  [_ z  /  x ]_ A  =  [_ v  /  x ]_ A
)
1511, 13, 143imtr4i 266 . . . 4  |-  ( ( z R w  /\  w R v )  -> 
z R v )
1615adantl 466 . . 3  |-  ( ( T.  /\  ( z R w  /\  w R v ) )  ->  z R v )
17 vex 3071 . . . . 5  |-  z  e. 
_V
18 eqid 2451 . . . . . 6  |-  [_ z  /  x ]_ A  = 
[_ z  /  x ]_ A
196, 1eqerlem 7233 . . . . . 6  |-  ( z R z  <->  [_ z  /  x ]_ A  =  [_ z  /  x ]_ A
)
2018, 19mpbir 209 . . . . 5  |-  z R z
2117, 202th 239 . . . 4  |-  ( z  e.  _V  <->  z R
z )
2221a1i 11 . . 3  |-  ( T. 
->  ( z  e.  _V  <->  z R z ) )
233, 10, 16, 22iserd 7227 . 2  |-  ( T. 
->  R  Er  _V )
2423trud 1379 1  |-  R  Er  _V
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1370   T. wtru 1371    e. wcel 1758   _Vcvv 3068   [_csb 3386   class class class wbr 4390   {copab 4447   Rel wrel 4943    Er wer 7198
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4511  ax-nul 4519  ax-pr 4629
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3070  df-sbc 3285  df-csb 3387  df-dif 3429  df-un 3431  df-in 3433  df-ss 3440  df-nul 3736  df-if 3890  df-sn 3976  df-pr 3978  df-op 3982  df-br 4391  df-opab 4449  df-xp 4944  df-rel 4945  df-cnv 4946  df-co 4947  df-dm 4948  df-er 7201
This theorem is referenced by:  ider  7235  frgpuplem  16373  fneer  28698
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