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Theorem resiexg 4997
Description: The existence of a restricted identity function, proved without using the Axiom of Replacement (unlike resfunexg 5737). (Contributed by NM, 13-Jan-2007.)
Assertion
Ref Expression
resiexg  |-  ( A  e.  V  ->  (  _I  |`  A )  e. 
_V )

Proof of Theorem resiexg
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 relres 4983 . . 3  |-  Rel  (  _I  |`  A )
2 simpr 447 . . . . 5  |-  ( ( x  =  y  /\  x  e.  A )  ->  x  e.  A )
3 eleq1 2343 . . . . . 6  |-  ( x  =  y  ->  (
x  e.  A  <->  y  e.  A ) )
43biimpa 470 . . . . 5  |-  ( ( x  =  y  /\  x  e.  A )  ->  y  e.  A )
52, 4jca 518 . . . 4  |-  ( ( x  =  y  /\  x  e.  A )  ->  ( x  e.  A  /\  y  e.  A
) )
6 vex 2791 . . . . . 6  |-  y  e. 
_V
76opelres 4960 . . . . 5  |-  ( <.
x ,  y >.  e.  (  _I  |`  A )  <-> 
( <. x ,  y
>.  e.  _I  /\  x  e.  A ) )
8 df-br 4024 . . . . . . 7  |-  ( x  _I  y  <->  <. x ,  y >.  e.  _I  )
96ideq 4836 . . . . . . 7  |-  ( x  _I  y  <->  x  =  y )
108, 9bitr3i 242 . . . . . 6  |-  ( <.
x ,  y >.  e.  _I  <->  x  =  y
)
1110anbi1i 676 . . . . 5  |-  ( (
<. x ,  y >.  e.  _I  /\  x  e.  A )  <->  ( x  =  y  /\  x  e.  A ) )
127, 11bitri 240 . . . 4  |-  ( <.
x ,  y >.  e.  (  _I  |`  A )  <-> 
( x  =  y  /\  x  e.  A
) )
13 opelxp 4719 . . . 4  |-  ( <.
x ,  y >.  e.  ( A  X.  A
)  <->  ( x  e.  A  /\  y  e.  A ) )
145, 12, 133imtr4i 257 . . 3  |-  ( <.
x ,  y >.  e.  (  _I  |`  A )  ->  <. x ,  y
>.  e.  ( A  X.  A ) )
151, 14relssi 4778 . 2  |-  (  _I  |`  A )  C_  ( A  X.  A )
16 xpexg 4800 . . 3  |-  ( ( A  e.  V  /\  A  e.  V )  ->  ( A  X.  A
)  e.  _V )
1716anidms 626 . 2  |-  ( A  e.  V  ->  ( A  X.  A )  e. 
_V )
18 ssexg 4160 . 2  |-  ( ( (  _I  |`  A ) 
C_  ( A  X.  A )  /\  ( A  X.  A )  e. 
_V )  ->  (  _I  |`  A )  e. 
_V )
1915, 17, 18sylancr 644 1  |-  ( A  e.  V  ->  (  _I  |`  A )  e. 
_V )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   _Vcvv 2788    C_ wss 3152   <.cop 3643   class class class wbr 4023    _I cid 4304    X. cxp 4687    |` cres 4691
This theorem is referenced by:  ordiso  7231  wdomref  7286  dfac9  7762  ndxarg  13168  idfu2nd  13751  idfu1st  13753  idfucl  13755  setcid  13918  pf1ind  19438  relexp0  23436  relexpsucr  23437  dispos  24699  infemb  25271  grphidmor2  25365  eldioph2lem1  26251  eldioph2lem2  26252  islinds2  26695  dib0  30727  dicn0  30755  cdlemn11a  30770  dihord6apre  30819  dihatlat  30897  dihpN  30899
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-id 4309  df-xp 4695  df-rel 4696  df-res 4701
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