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Theorem ereq1 6179
Description: Equality theorem for equivalence predicate. (Contributed by NM, 4-Jun-1995.) (Revised by Mario Carneiro, 12-Aug-2015.)
Assertion
Ref Expression
ereq1  |-  ( R  =  S  ->  ( R  Er  A  <->  S  Er  A ) )

Proof of Theorem ereq1
StepHypRef Expression
1 releq 4448 . . 3  |-  ( R  =  S  ->  ( Rel  R  <->  Rel  S ) )
2 dmeq 4563 . . . 4  |-  ( R  =  S  ->  dom  R  =  dom  S )
32eqeq1d 2090 . . 3  |-  ( R  =  S  ->  ( dom  R  =  A  <->  dom  S  =  A ) )
4 cnveq 4537 . . . . . 6  |-  ( R  =  S  ->  `' R  =  `' S
)
5 coeq1 4521 . . . . . . 7  |-  ( R  =  S  ->  ( R  o.  R )  =  ( S  o.  R ) )
6 coeq2 4522 . . . . . . 7  |-  ( R  =  S  ->  ( S  o.  R )  =  ( S  o.  S ) )
75, 6eqtrd 2114 . . . . . 6  |-  ( R  =  S  ->  ( R  o.  R )  =  ( S  o.  S ) )
84, 7uneq12d 3128 . . . . 5  |-  ( R  =  S  ->  ( `' R  u.  ( R  o.  R )
)  =  ( `' S  u.  ( S  o.  S ) ) )
98sseq1d 3027 . . . 4  |-  ( R  =  S  ->  (
( `' R  u.  ( R  o.  R
) )  C_  R  <->  ( `' S  u.  ( S  o.  S )
)  C_  R )
)
10 sseq2 3022 . . . 4  |-  ( R  =  S  ->  (
( `' S  u.  ( S  o.  S
) )  C_  R  <->  ( `' S  u.  ( S  o.  S )
)  C_  S )
)
119, 10bitrd 186 . . 3  |-  ( R  =  S  ->  (
( `' R  u.  ( R  o.  R
) )  C_  R  <->  ( `' S  u.  ( S  o.  S )
)  C_  S )
)
121, 3, 113anbi123d 1244 . 2  |-  ( R  =  S  ->  (
( Rel  R  /\  dom  R  =  A  /\  ( `' R  u.  ( R  o.  R )
)  C_  R )  <->  ( Rel  S  /\  dom  S  =  A  /\  ( `' S  u.  ( S  o.  S )
)  C_  S )
) )
13 df-er 6172 . 2  |-  ( R  Er  A  <->  ( Rel  R  /\  dom  R  =  A  /\  ( `' R  u.  ( R  o.  R ) ) 
C_  R ) )
14 df-er 6172 . 2  |-  ( S  Er  A  <->  ( Rel  S  /\  dom  S  =  A  /\  ( `' S  u.  ( S  o.  S ) ) 
C_  S ) )
1512, 13, 143bitr4g 221 1  |-  ( R  =  S  ->  ( R  Er  A  <->  S  Er  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 103    /\ w3a 920    = wceq 1285    u. cun 2972    C_ wss 2974   `'ccnv 4370   dom cdm 4371    o. ccom 4375   Rel wrel 4376    Er wer 6169
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-v 2604  df-un 2978  df-in 2980  df-ss 2987  df-sn 3412  df-pr 3413  df-op 3415  df-br 3794  df-opab 3848  df-rel 4378  df-cnv 4379  df-co 4380  df-dm 4381  df-er 6172
This theorem is referenced by:  riinerm  6245
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