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Theorem dmresi 5272
Description: The domain of a restricted identity function. (Contributed by NM, 27-Aug-2004.)
Assertion
Ref Expression
dmresi  |-  dom  (  _I  |`  A )  =  A

Proof of Theorem dmresi
StepHypRef Expression
1 ssv 3487 . . 3  |-  A  C_  _V
2 dmi 5165 . . 3  |-  dom  _I  =  _V
31, 2sseqtr4i 3500 . 2  |-  A  C_  dom  _I
4 ssdmres 5243 . 2  |-  ( A 
C_  dom  _I  <->  dom  (  _I  |`  A )  =  A )
53, 4mpbi 208 1  |-  dom  (  _I  |`  A )  =  A
Colors of variables: wff setvar class
Syntax hints:    = wceq 1370   _Vcvv 3078    C_ wss 3439    _I cid 4742   dom cdm 4951    |` cres 4953
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pr 4642
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-sn 3989  df-pr 3991  df-op 3995  df-br 4404  df-opab 4462  df-id 4747  df-xp 4957  df-rel 4958  df-dm 4961  df-res 4963
This theorem is referenced by:  fnresi  5639  iordsmo  6931  hartogslem1  7871  dfac9  8420  hsmexlem5  8714  dirdm  15527  islinds2  18377  lindsind2  18383  f1linds  18389  wilthlem3  22551  ausisusgra  23458  cusgraexilem2  23554  relexpdm  27504  filnetlem3  28772  filnetlem4  28773
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