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Theorem ereq1 7369
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 4928 . . 3  |-  ( R  =  S  ->  ( Rel  R  <->  Rel  S ) )
2 dmeq 5046 . . . 4  |-  ( R  =  S  ->  dom  R  =  dom  S )
32eqeq1d 2422 . . 3  |-  ( R  =  S  ->  ( dom  R  =  A  <->  dom  S  =  A ) )
4 cnveq 5019 . . . . . 6  |-  ( R  =  S  ->  `' R  =  `' S
)
5 coeq1 5003 . . . . . . 7  |-  ( R  =  S  ->  ( R  o.  R )  =  ( S  o.  R ) )
6 coeq2 5004 . . . . . . 7  |-  ( R  =  S  ->  ( S  o.  R )  =  ( S  o.  S ) )
75, 6eqtrd 2461 . . . . . 6  |-  ( R  =  S  ->  ( R  o.  R )  =  ( S  o.  S ) )
84, 7uneq12d 3618 . . . . 5  |-  ( R  =  S  ->  ( `' R  u.  ( R  o.  R )
)  =  ( `' S  u.  ( S  o.  S ) ) )
98sseq1d 3488 . . . 4  |-  ( R  =  S  ->  (
( `' R  u.  ( R  o.  R
) )  C_  R  <->  ( `' S  u.  ( S  o.  S )
)  C_  R )
)
10 sseq2 3483 . . . 4  |-  ( R  =  S  ->  (
( `' S  u.  ( S  o.  S
) )  C_  R  <->  ( `' S  u.  ( S  o.  S )
)  C_  S )
)
119, 10bitrd 256 . . 3  |-  ( R  =  S  ->  (
( `' R  u.  ( R  o.  R
) )  C_  R  <->  ( `' S  u.  ( S  o.  S )
)  C_  S )
)
121, 3, 113anbi123d 1335 . 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 7362 . 2  |-  ( R  Er  A  <->  ( Rel  R  /\  dom  R  =  A  /\  ( `' R  u.  ( R  o.  R ) ) 
C_  R ) )
14 df-er 7362 . 2  |-  ( S  Er  A  <->  ( Rel  S  /\  dom  S  =  A  /\  ( `' S  u.  ( S  o.  S ) ) 
C_  S ) )
1512, 13, 143bitr4g 291 1  |-  ( R  =  S  ->  ( R  Er  A  <->  S  Er  A ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ w3a 982    = wceq 1437    u. cun 3431    C_ wss 3433   `'ccnv 4844   dom cdm 4845    o. ccom 4849   Rel wrel 4850    Er wer 7359
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-rab 2782  df-v 3080  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-nul 3759  df-if 3907  df-sn 3994  df-pr 3996  df-op 4000  df-br 4418  df-opab 4476  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-er 7362
This theorem is referenced by:  riiner  7435  efglem  17294  efger  17296  efgrelexlemb  17328  efgcpbllemb  17333  frgpuplem  17350  qtophaus  28499  pstmxmet  28536
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