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Theorem ereq1 6542
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 4709 . . 3  |-  ( R  =  S  ->  ( Rel  R  <->  Rel  S ) )
2 dmeq 4828 . . . 4  |-  ( R  =  S  ->  dom  R  =  dom  S )
32eqeq1d 2186 . . 3  |-  ( R  =  S  ->  ( dom  R  =  A  <->  dom  S  =  A ) )
4 cnveq 4802 . . . . . 6  |-  ( R  =  S  ->  `' R  =  `' S
)
5 coeq1 4785 . . . . . . 7  |-  ( R  =  S  ->  ( R  o.  R )  =  ( S  o.  R ) )
6 coeq2 4786 . . . . . . 7  |-  ( R  =  S  ->  ( S  o.  R )  =  ( S  o.  S ) )
75, 6eqtrd 2210 . . . . . 6  |-  ( R  =  S  ->  ( R  o.  R )  =  ( S  o.  S ) )
84, 7uneq12d 3291 . . . . 5  |-  ( R  =  S  ->  ( `' R  u.  ( R  o.  R )
)  =  ( `' S  u.  ( S  o.  S ) ) )
98sseq1d 3185 . . . 4  |-  ( R  =  S  ->  (
( `' R  u.  ( R  o.  R
) )  C_  R  <->  ( `' S  u.  ( S  o.  S )
)  C_  R )
)
10 sseq2 3180 . . . 4  |-  ( R  =  S  ->  (
( `' S  u.  ( S  o.  S
) )  C_  R  <->  ( `' S  u.  ( S  o.  S )
)  C_  S )
)
119, 10bitrd 188 . . 3  |-  ( R  =  S  ->  (
( `' R  u.  ( R  o.  R
) )  C_  R  <->  ( `' S  u.  ( S  o.  S )
)  C_  S )
)
121, 3, 113anbi123d 1312 . 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 6535 . 2  |-  ( R  Er  A  <->  ( Rel  R  /\  dom  R  =  A  /\  ( `' R  u.  ( R  o.  R ) ) 
C_  R ) )
14 df-er 6535 . 2  |-  ( S  Er  A  <->  ( Rel  S  /\  dom  S  =  A  /\  ( `' S  u.  ( S  o.  S ) ) 
C_  S ) )
1512, 13, 143bitr4g 223 1  |-  ( R  =  S  ->  ( R  Er  A  <->  S  Er  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 105    /\ w3a 978    = wceq 1353    u. cun 3128    C_ wss 3130   `'ccnv 4626   dom cdm 4627    o. ccom 4631   Rel wrel 4632    Er wer 6532
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2740  df-un 3134  df-in 3136  df-ss 3143  df-sn 3599  df-pr 3600  df-op 3602  df-br 4005  df-opab 4066  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-er 6535
This theorem is referenced by:  riinerm  6608
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