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Theorem resieq 4981
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 4043 . . . . 5  |-  ( x  =  C  ->  ( B (  _I  |`  A ) x  <->  B (  _I  |`  A ) C ) )
2 eqeq2 2305 . . . . 5  |-  ( x  =  C  ->  ( B  =  x  <->  B  =  C ) )
31, 2bibi12d 312 . . . 4  |-  ( x  =  C  ->  (
( B (  _I  |`  A ) x  <->  B  =  x )  <->  ( B
(  _I  |`  A ) C  <->  B  =  C
) ) )
43imbi2d 307 . . 3  |-  ( x  =  C  ->  (
( B  e.  A  ->  ( B (  _I  |`  A ) x  <->  B  =  x ) )  <->  ( B  e.  A  ->  ( B (  _I  |`  A ) C  <->  B  =  C
) ) ) )
5 vex 2804 . . . . 5  |-  x  e. 
_V
65opres 4980 . . . 4  |-  ( B  e.  A  ->  ( <. B ,  x >.  e.  (  _I  |`  A )  <->  <. B ,  x >.  e.  _I  ) )
7 df-br 4040 . . . 4  |-  ( B (  _I  |`  A ) x  <->  <. B ,  x >.  e.  (  _I  |`  A ) )
85ideq 4852 . . . . 5  |-  ( B  _I  x  <->  B  =  x )
9 df-br 4040 . . . . 5  |-  ( B  _I  x  <->  <. B ,  x >.  e.  _I  )
108, 9bitr3i 242 . . . 4  |-  ( B  =  x  <->  <. B ,  x >.  e.  _I  )
116, 7, 103bitr4g 279 . . 3  |-  ( B  e.  A  ->  ( B (  _I  |`  A ) x  <->  B  =  x
) )
124, 11vtoclg 2856 . 2  |-  ( C  e.  A  ->  ( B  e.  A  ->  ( B (  _I  |`  A ) C  <->  B  =  C
) ) )
1312impcom 419 1  |-  ( ( B  e.  A  /\  C  e.  A )  ->  ( B (  _I  |`  A ) C  <->  B  =  C ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696   <.cop 3656   class class class wbr 4039    _I cid 4320    |` cres 4707
This theorem is referenced by:  foeqcnvco  5820  f1eqcocnv  5821  dfle2  10497  pospo  14123  dirref  14373
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-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
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-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  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-id 4325  df-xp 4711  df-rel 4712  df-res 4717
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