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Theorem ereq2 7212
Description: Equality theorem for equivalence predicate. (Contributed by Mario Carneiro, 12-Aug-2015.)
Assertion
Ref Expression
ereq2  |-  ( A  =  B  ->  ( R  Er  A  <->  R  Er  B ) )

Proof of Theorem ereq2
StepHypRef Expression
1 eqeq2 2466 . . 3  |-  ( A  =  B  ->  ( dom  R  =  A  <->  dom  R  =  B ) )
213anbi2d 1295 . 2  |-  ( A  =  B  ->  (
( Rel  R  /\  dom  R  =  A  /\  ( `' R  u.  ( R  o.  R )
)  C_  R )  <->  ( Rel  R  /\  dom  R  =  B  /\  ( `' R  u.  ( R  o.  R )
)  C_  R )
) )
3 df-er 7204 . 2  |-  ( R  Er  A  <->  ( Rel  R  /\  dom  R  =  A  /\  ( `' R  u.  ( R  o.  R ) ) 
C_  R ) )
4 df-er 7204 . 2  |-  ( R  Er  B  <->  ( Rel  R  /\  dom  R  =  B  /\  ( `' R  u.  ( R  o.  R ) ) 
C_  R ) )
52, 3, 43bitr4g 288 1  |-  ( A  =  B  ->  ( R  Er  A  <->  R  Er  B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ w3a 965    = wceq 1370    u. cun 3427    C_ wss 3429   `'ccnv 4940   dom cdm 4941    o. ccom 4945   Rel wrel 4946    Er wer 7201
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-ext 2430
This theorem depends on definitions:  df-bi 185  df-an 371  df-3an 967  df-cleq 2443  df-er 7204
This theorem is referenced by:  iserd  7230  efgval  16327  frgp0  16370  frgpmhm  16375
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