MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ereq1 Structured version   Visualization version   Unicode version

Theorem ereq1 7388
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 4922 . . 3  |-  ( R  =  S  ->  ( Rel  R  <->  Rel  S ) )
2 dmeq 5040 . . . 4  |-  ( R  =  S  ->  dom  R  =  dom  S )
32eqeq1d 2473 . . 3  |-  ( R  =  S  ->  ( dom  R  =  A  <->  dom  S  =  A ) )
4 cnveq 5013 . . . . . 6  |-  ( R  =  S  ->  `' R  =  `' S
)
5 coeq1 4997 . . . . . . 7  |-  ( R  =  S  ->  ( R  o.  R )  =  ( S  o.  R ) )
6 coeq2 4998 . . . . . . 7  |-  ( R  =  S  ->  ( S  o.  R )  =  ( S  o.  S ) )
75, 6eqtrd 2505 . . . . . 6  |-  ( R  =  S  ->  ( R  o.  R )  =  ( S  o.  S ) )
84, 7uneq12d 3580 . . . . 5  |-  ( R  =  S  ->  ( `' R  u.  ( R  o.  R )
)  =  ( `' S  u.  ( S  o.  S ) ) )
98sseq1d 3445 . . . 4  |-  ( R  =  S  ->  (
( `' R  u.  ( R  o.  R
) )  C_  R  <->  ( `' S  u.  ( S  o.  S )
)  C_  R )
)
10 sseq2 3440 . . . 4  |-  ( R  =  S  ->  (
( `' S  u.  ( S  o.  S
) )  C_  R  <->  ( `' S  u.  ( S  o.  S )
)  C_  S )
)
119, 10bitrd 261 . . 3  |-  ( R  =  S  ->  (
( `' R  u.  ( R  o.  R
) )  C_  R  <->  ( `' S  u.  ( S  o.  S )
)  C_  S )
)
121, 3, 113anbi123d 1365 . 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 7381 . 2  |-  ( R  Er  A  <->  ( Rel  R  /\  dom  R  =  A  /\  ( `' R  u.  ( R  o.  R ) ) 
C_  R ) )
14 df-er 7381 . 2  |-  ( S  Er  A  <->  ( Rel  S  /\  dom  S  =  A  /\  ( `' S  u.  ( S  o.  S ) ) 
C_  S ) )
1512, 13, 143bitr4g 296 1  |-  ( R  =  S  ->  ( R  Er  A  <->  S  Er  A ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    /\ w3a 1007    = wceq 1452    u. cun 3388    C_ wss 3390   `'ccnv 4838   dom cdm 4839    o. ccom 4843   Rel wrel 4844    Er wer 7378
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-rab 2765  df-v 3033  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-op 3966  df-br 4396  df-opab 4455  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-er 7381
This theorem is referenced by:  riiner  7454  efglem  17444  efger  17446  efgrelexlemb  17478  efgcpbllemb  17483  frgpuplem  17500  qtophaus  28737  pstmxmet  28774
  Copyright terms: Public domain W3C validator