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Theorem resieq 5140
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 4315 . . . . 5  |-  ( x  =  C  ->  ( B (  _I  |`  A ) x  <->  B (  _I  |`  A ) C ) )
2 eqeq2 2452 . . . . 5  |-  ( x  =  C  ->  ( B  =  x  <->  B  =  C ) )
31, 2bibi12d 321 . . . 4  |-  ( x  =  C  ->  (
( B (  _I  |`  A ) x  <->  B  =  x )  <->  ( B
(  _I  |`  A ) C  <->  B  =  C
) ) )
43imbi2d 316 . . 3  |-  ( x  =  C  ->  (
( B  e.  A  ->  ( B (  _I  |`  A ) x  <->  B  =  x ) )  <->  ( B  e.  A  ->  ( B (  _I  |`  A ) C  <->  B  =  C
) ) ) )
5 vex 2994 . . . . 5  |-  x  e. 
_V
65opres 5139 . . . 4  |-  ( B  e.  A  ->  ( <. B ,  x >.  e.  (  _I  |`  A )  <->  <. B ,  x >.  e.  _I  ) )
7 df-br 4312 . . . 4  |-  ( B (  _I  |`  A ) x  <->  <. B ,  x >.  e.  (  _I  |`  A ) )
85ideq 5011 . . . . 5  |-  ( B  _I  x  <->  B  =  x )
9 df-br 4312 . . . . 5  |-  ( B  _I  x  <->  <. B ,  x >.  e.  _I  )
108, 9bitr3i 251 . . . 4  |-  ( B  =  x  <->  <. B ,  x >.  e.  _I  )
116, 7, 103bitr4g 288 . . 3  |-  ( B  e.  A  ->  ( B (  _I  |`  A ) x  <->  B  =  x
) )
124, 11vtoclg 3049 . 2  |-  ( C  e.  A  ->  ( B  e.  A  ->  ( B (  _I  |`  A ) C  <->  B  =  C
) ) )
1312impcom 430 1  |-  ( ( B  e.  A  /\  C  e.  A )  ->  ( B (  _I  |`  A ) C  <->  B  =  C ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756   <.cop 3902   class class class wbr 4311    _I cid 4650    |` cres 4861
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4432  ax-nul 4440  ax-pr 4550
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-ral 2739  df-rex 2740  df-rab 2743  df-v 2993  df-dif 3350  df-un 3352  df-in 3354  df-ss 3361  df-nul 3657  df-if 3811  df-sn 3897  df-pr 3899  df-op 3903  df-br 4312  df-opab 4370  df-id 4655  df-xp 4865  df-rel 4866  df-res 4871
This theorem is referenced by:  foeqcnvco  6017  f1eqcocnv  6018  dfle2  11143  pospo  15162  dirref  15424  ustref  19812  trust  19823
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