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Theorem resieq 5148
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 4208 . . . . 5  |-  ( x  =  C  ->  ( B (  _I  |`  A ) x  <->  B (  _I  |`  A ) C ) )
2 eqeq2 2444 . . . . 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 2951 . . . . 5  |-  x  e. 
_V
65opres 5147 . . . 4  |-  ( B  e.  A  ->  ( <. B ,  x >.  e.  (  _I  |`  A )  <->  <. B ,  x >.  e.  _I  ) )
7 df-br 4205 . . . 4  |-  ( B (  _I  |`  A ) x  <->  <. B ,  x >.  e.  (  _I  |`  A ) )
85ideq 5017 . . . . 5  |-  ( B  _I  x  <->  B  =  x )
9 df-br 4205 . . . . 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 3003 . 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 1652    e. wcel 1725   <.cop 3809   class class class wbr 4204    _I cid 4485    |` cres 4872
This theorem is referenced by:  foeqcnvco  6019  f1eqcocnv  6020  dfle2  10732  pospo  14422  dirref  14672  ustref  18240  trust  18251
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-br 4205  df-opab 4259  df-id 4490  df-xp 4876  df-rel 4877  df-res 4882
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