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Theorem resiexg 5013
Description: The existence of a restricted identity function, proved without using the Axiom of Replacement (unlike resfunexg 5753). (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 4999 . . 3  |-  Rel  (  _I  |`  A )
2 simpr 447 . . . . 5  |-  ( ( x  =  y  /\  x  e.  A )  ->  x  e.  A )
3 eleq1 2356 . . . . . 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 2804 . . . . . 6  |-  y  e. 
_V
76opelres 4976 . . . . 5  |-  ( <.
x ,  y >.  e.  (  _I  |`  A )  <-> 
( <. x ,  y
>.  e.  _I  /\  x  e.  A ) )
8 df-br 4040 . . . . . . 7  |-  ( x  _I  y  <->  <. x ,  y >.  e.  _I  )
96ideq 4852 . . . . . . 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 4735 . . . 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 4794 . 2  |-  (  _I  |`  A )  C_  ( A  X.  A )
16 xpexg 4816 . . 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 4176 . 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 1632    e. wcel 1696   _Vcvv 2801    C_ wss 3165   <.cop 3656   class class class wbr 4039    _I cid 4320    X. cxp 4703    |` cres 4707
This theorem is referenced by:  ordiso  7247  wdomref  7302  dfac9  7778  ndxarg  13184  idfu2nd  13767  idfu1st  13769  idfucl  13771  setcid  13934  pf1ind  19454  relexp0  24040  relexpsucr  24041  dispos  25390  infemb  25962  grphidmor2  26056  eldioph2lem1  26942  eldioph2lem2  26943  islinds2  27386  dib0  31976  dicn0  32004  cdlemn11a  32019  dihord6apre  32068  dihatlat  32146  dihpN  32148
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-id 4325  df-xp 4711  df-rel 4712  df-res 4717
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