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Theorem ereq1 6683
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 4787 . . 3  |-  ( R  =  S  ->  ( Rel  R  <->  Rel  S ) )
2 dmeq 4895 . . . 4  |-  ( R  =  S  ->  dom  R  =  dom  S )
32eqeq1d 2304 . . 3  |-  ( R  =  S  ->  ( dom  R  =  A  <->  dom  S  =  A ) )
4 cnveq 4871 . . . . . 6  |-  ( R  =  S  ->  `' R  =  `' S
)
5 coeq1 4857 . . . . . . 7  |-  ( R  =  S  ->  ( R  o.  R )  =  ( S  o.  R ) )
6 coeq2 4858 . . . . . . 7  |-  ( R  =  S  ->  ( S  o.  R )  =  ( S  o.  S ) )
75, 6eqtrd 2328 . . . . . 6  |-  ( R  =  S  ->  ( R  o.  R )  =  ( S  o.  S ) )
84, 7uneq12d 3343 . . . . 5  |-  ( R  =  S  ->  ( `' R  u.  ( R  o.  R )
)  =  ( `' S  u.  ( S  o.  S ) ) )
98sseq1d 3218 . . . 4  |-  ( R  =  S  ->  (
( `' R  u.  ( R  o.  R
) )  C_  R  <->  ( `' S  u.  ( S  o.  S )
)  C_  R )
)
10 sseq2 3213 . . . 4  |-  ( R  =  S  ->  (
( `' S  u.  ( S  o.  S
) )  C_  R  <->  ( `' S  u.  ( S  o.  S )
)  C_  S )
)
119, 10bitrd 244 . . 3  |-  ( R  =  S  ->  (
( `' R  u.  ( R  o.  R
) )  C_  R  <->  ( `' S  u.  ( S  o.  S )
)  C_  S )
)
121, 3, 113anbi123d 1252 . 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 6676 . 2  |-  ( R  Er  A  <->  ( Rel  R  /\  dom  R  =  A  /\  ( `' R  u.  ( R  o.  R ) ) 
C_  R ) )
14 df-er 6676 . 2  |-  ( S  Er  A  <->  ( Rel  S  /\  dom  S  =  A  /\  ( `' S  u.  ( S  o.  S ) ) 
C_  S ) )
1512, 13, 143bitr4g 279 1  |-  ( R  =  S  ->  ( R  Er  A  <->  S  Er  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ w3a 934    = wceq 1632    u. cun 3163    C_ wss 3165   `'ccnv 4704   dom cdm 4705    o. ccom 4709   Rel wrel 4710    Er wer 6673
This theorem is referenced by:  riiner  6748  efglem  15041  efger  15043  efgrelexlemb  15075  efgcpbllemb  15080  frgpuplem  15097  isibcg  26294
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-br 4040  df-opab 4094  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-er 6676
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