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Theorem eqer 7336
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 5116 . . . 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 2462 . . . . 5  |-  ( [_ z  /  x ]_ A  =  [_ w  /  x ]_ A  ->  [_ w  /  x ]_ A  = 
[_ z  /  x ]_ A )
6 eqer.1 . . . . . 6  |-  ( x  =  y  ->  A  =  B )
76, 1eqerlem 7335 . . . . 5  |-  ( z R w  <->  [_ z  /  x ]_ A  =  [_ w  /  x ]_ A
)
86, 1eqerlem 7335 . . . . 5  |-  ( w R z  <->  [_ w  /  x ]_ A  =  [_ z  /  x ]_ A
)
95, 7, 83imtr4i 266 . . . 4  |-  ( z R w  ->  w R z )
109adantl 464 . . 3  |-  ( ( T.  /\  z R w )  ->  w R z )
11 eqtr 2480 . . . . 5  |-  ( (
[_ z  /  x ]_ A  =  [_ w  /  x ]_ A  /\  [_ w  /  x ]_ A  =  [_ v  /  x ]_ A )  ->  [_ z  /  x ]_ A  =  [_ v  /  x ]_ A )
126, 1eqerlem 7335 . . . . . 6  |-  ( w R v  <->  [_ w  /  x ]_ A  =  [_ v  /  x ]_ A
)
137, 12anbi12i 695 . . . . 5  |-  ( ( z R w  /\  w R v )  <->  ( [_ z  /  x ]_ A  =  [_ w  /  x ]_ A  /\  [_ w  /  x ]_ A  = 
[_ v  /  x ]_ A ) )
146, 1eqerlem 7335 . . . . 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 464 . . 3  |-  ( ( T.  /\  ( z R w  /\  w R v ) )  ->  z R v )
17 vex 3109 . . . . 5  |-  z  e. 
_V
18 eqid 2454 . . . . . 6  |-  [_ z  /  x ]_ A  = 
[_ z  /  x ]_ A
196, 1eqerlem 7335 . . . . . 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 7329 . 2  |-  ( T. 
->  R  Er  _V )
2423trud 1407 1  |-  R  Er  _V
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1398   T. wtru 1399    e. wcel 1823   _Vcvv 3106   [_csb 3420   class class class wbr 4439   {copab 4496   Rel wrel 4993    Er wer 7300
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-br 4440  df-opab 4498  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-er 7303
This theorem is referenced by:  ider  7337  frgpuplem  16992  fneer  30414
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