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Theorem resieq 4693
Description: A restricted identity relation is equivalent to equality in its domain. (Contributed by NM, 30-Apr-2004.)
Assertion
Ref Expression
resieq  |-  ( ( B  e.  A  /\  C  e.  A )  ->  ( B (  _I  |`  A ) C  <->  B  =  C ) )

Proof of Theorem resieq
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 breq2 3826 . . . . 5  |-  ( x  =  C  ->  ( B (  _I  |`  A ) x  <->  B (  _I  |`  A ) C ) )
2 eqeq2 2094 . . . . 5  |-  ( x  =  C  ->  ( B  =  x  <->  B  =  C ) )
31, 2bibi12d 233 . . . 4  |-  ( x  =  C  ->  (
( B (  _I  |`  A ) x  <->  B  =  x )  <->  ( B
(  _I  |`  A ) C  <->  B  =  C
) ) )
43imbi2d 228 . . 3  |-  ( x  =  C  ->  (
( B  e.  A  ->  ( B (  _I  |`  A ) x  <->  B  =  x ) )  <->  ( B  e.  A  ->  ( B (  _I  |`  A ) C  <->  B  =  C
) ) ) )
5 vex 2618 . . . . 5  |-  x  e. 
_V
65opres 4692 . . . 4  |-  ( B  e.  A  ->  ( <. B ,  x >.  e.  (  _I  |`  A )  <->  <. B ,  x >.  e.  _I  ) )
7 df-br 3823 . . . 4  |-  ( B (  _I  |`  A ) x  <->  <. B ,  x >.  e.  (  _I  |`  A ) )
85ideq 4558 . . . . 5  |-  ( B  _I  x  <->  B  =  x )
9 df-br 3823 . . . . 5  |-  ( B  _I  x  <->  <. B ,  x >.  e.  _I  )
108, 9bitr3i 184 . . . 4  |-  ( B  =  x  <->  <. B ,  x >.  e.  _I  )
116, 7, 103bitr4g 221 . . 3  |-  ( B  e.  A  ->  ( B (  _I  |`  A ) x  <->  B  =  x
) )
124, 11vtoclg 2672 . 2  |-  ( C  e.  A  ->  ( B  e.  A  ->  ( B (  _I  |`  A ) C  <->  B  =  C
) ) )
1312impcom 123 1  |-  ( ( B  e.  A  /\  C  e.  A )  ->  ( B (  _I  |`  A ) C  <->  B  =  C ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    = wceq 1287    e. wcel 1436   <.cop 3434   class class class wbr 3822    _I cid 4091    |` cres 4415
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 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-sep 3934  ax-pow 3986  ax-pr 4012
This theorem depends on definitions:  df-bi 115  df-3an 924  df-tru 1290  df-nf 1393  df-sb 1690  df-eu 1948  df-mo 1949  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ral 2360  df-rex 2361  df-v 2617  df-un 2992  df-in 2994  df-ss 3001  df-pw 3417  df-sn 3437  df-pr 3438  df-op 3440  df-br 3823  df-opab 3877  df-id 4096  df-xp 4419  df-rel 4420  df-res 4425
This theorem is referenced by:  foeqcnvco  5532  f1eqcocnv  5533
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