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Theorem resieq 5089
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 4150 . . . . 5  |-  ( x  =  C  ->  ( B (  _I  |`  A ) x  <->  B (  _I  |`  A ) C ) )
2 eqeq2 2389 . . . . 5  |-  ( x  =  C  ->  ( B  =  x  <->  B  =  C ) )
31, 2bibi12d 313 . . . 4  |-  ( x  =  C  ->  (
( B (  _I  |`  A ) x  <->  B  =  x )  <->  ( B
(  _I  |`  A ) C  <->  B  =  C
) ) )
43imbi2d 308 . . 3  |-  ( x  =  C  ->  (
( B  e.  A  ->  ( B (  _I  |`  A ) x  <->  B  =  x ) )  <->  ( B  e.  A  ->  ( B (  _I  |`  A ) C  <->  B  =  C
) ) ) )
5 vex 2895 . . . . 5  |-  x  e. 
_V
65opres 5088 . . . 4  |-  ( B  e.  A  ->  ( <. B ,  x >.  e.  (  _I  |`  A )  <->  <. B ,  x >.  e.  _I  ) )
7 df-br 4147 . . . 4  |-  ( B (  _I  |`  A ) x  <->  <. B ,  x >.  e.  (  _I  |`  A ) )
85ideq 4958 . . . . 5  |-  ( B  _I  x  <->  B  =  x )
9 df-br 4147 . . . . 5  |-  ( B  _I  x  <->  <. B ,  x >.  e.  _I  )
108, 9bitr3i 243 . . . 4  |-  ( B  =  x  <->  <. B ,  x >.  e.  _I  )
116, 7, 103bitr4g 280 . . 3  |-  ( B  e.  A  ->  ( B (  _I  |`  A ) x  <->  B  =  x
) )
124, 11vtoclg 2947 . 2  |-  ( C  e.  A  ->  ( B  e.  A  ->  ( B (  _I  |`  A ) C  <->  B  =  C
) ) )
1312impcom 420 1  |-  ( ( B  e.  A  /\  C  e.  A )  ->  ( B (  _I  |`  A ) C  <->  B  =  C ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1717   <.cop 3753   class class class wbr 4146    _I cid 4427    |` cres 4813
This theorem is referenced by:  foeqcnvco  5959  f1eqcocnv  5960  dfle2  10665  pospo  14350  dirref  14600  ustref  18162  trust  18173
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-sep 4264  ax-nul 4272  ax-pr 4337
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-ral 2647  df-rex 2648  df-rab 2651  df-v 2894  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-sn 3756  df-pr 3757  df-op 3759  df-br 4147  df-opab 4201  df-id 4432  df-xp 4817  df-rel 4818  df-res 4823
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