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Theorem resiexg 6513
Description: The existence of a restricted identity function, proved without using the Axiom of Replacement (unlike resfunexg 5942). (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 5137 . . 3  |-  Rel  (  _I  |`  A )
2 simpr 461 . . . . 5  |-  ( ( x  =  y  /\  x  e.  A )  ->  x  e.  A )
3 eleq1 2502 . . . . . 6  |-  ( x  =  y  ->  (
x  e.  A  <->  y  e.  A ) )
43biimpa 484 . . . . 5  |-  ( ( x  =  y  /\  x  e.  A )  ->  y  e.  A )
52, 4jca 532 . . . 4  |-  ( ( x  =  y  /\  x  e.  A )  ->  ( x  e.  A  /\  y  e.  A
) )
6 vex 2974 . . . . . 6  |-  y  e. 
_V
76opelres 5115 . . . . 5  |-  ( <.
x ,  y >.  e.  (  _I  |`  A )  <-> 
( <. x ,  y
>.  e.  _I  /\  x  e.  A ) )
8 df-br 4292 . . . . . . 7  |-  ( x  _I  y  <->  <. x ,  y >.  e.  _I  )
96ideq 4991 . . . . . . 7  |-  ( x  _I  y  <->  x  =  y )
108, 9bitr3i 251 . . . . . 6  |-  ( <.
x ,  y >.  e.  _I  <->  x  =  y
)
1110anbi1i 695 . . . . 5  |-  ( (
<. x ,  y >.  e.  _I  /\  x  e.  A )  <->  ( x  =  y  /\  x  e.  A ) )
127, 11bitri 249 . . . 4  |-  ( <.
x ,  y >.  e.  (  _I  |`  A )  <-> 
( x  =  y  /\  x  e.  A
) )
13 opelxp 4868 . . . 4  |-  ( <.
x ,  y >.  e.  ( A  X.  A
)  <->  ( x  e.  A  /\  y  e.  A ) )
145, 12, 133imtr4i 266 . . 3  |-  ( <.
x ,  y >.  e.  (  _I  |`  A )  ->  <. x ,  y
>.  e.  ( A  X.  A ) )
151, 14relssi 4930 . 2  |-  (  _I  |`  A )  C_  ( A  X.  A )
16 xpexg 6506 . . 3  |-  ( ( A  e.  V  /\  A  e.  V )  ->  ( A  X.  A
)  e.  _V )
1716anidms 645 . 2  |-  ( A  e.  V  ->  ( A  X.  A )  e. 
_V )
18 ssexg 4437 . 2  |-  ( ( (  _I  |`  A ) 
C_  ( A  X.  A )  /\  ( A  X.  A )  e. 
_V )  ->  (  _I  |`  A )  e. 
_V )
1915, 17, 18sylancr 663 1  |-  ( A  e.  V  ->  (  _I  |`  A )  e. 
_V )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    e. wcel 1756   _Vcvv 2971    C_ wss 3327   <.cop 3882   class class class wbr 4291    _I cid 4630    X. cxp 4837    |` cres 4841
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-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4412  ax-nul 4420  ax-pow 4469  ax-pr 4530  ax-un 6371
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 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2719  df-rex 2720  df-rab 2723  df-v 2973  df-dif 3330  df-un 3332  df-in 3334  df-ss 3341  df-nul 3637  df-if 3791  df-pw 3861  df-sn 3877  df-pr 3879  df-op 3883  df-uni 4091  df-br 4292  df-opab 4350  df-id 4635  df-xp 4845  df-rel 4846  df-res 4851
This theorem is referenced by:  ordiso  7729  wdomref  7786  dfac9  8304  ndxarg  14193  idfu2nd  14786  idfu1st  14788  idfucl  14790  setcid  14953  pf1ind  17788  islinds2  18241  ausisusgra  23278  cusgraexilem1  23373  sizeusglecusg  23393  relexp0  27330  relexpsucr  27331  eldioph2lem1  29096  eldioph2lem2  29097  dib0  34807  dicn0  34835  cdlemn11a  34850  dihord6apre  34899  dihatlat  34977  dihpN  34979
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