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Theorem resiexg 5121
Description: The existence of a restricted identity function, proved without using the Axiom of Replacement (unlike resfunexg 5889). (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 5107 . . 3  |-  Rel  (  _I  |`  A )
2 simpr 448 . . . . 5  |-  ( ( x  =  y  /\  x  e.  A )  ->  x  e.  A )
3 eleq1 2440 . . . . . 6  |-  ( x  =  y  ->  (
x  e.  A  <->  y  e.  A ) )
43biimpa 471 . . . . 5  |-  ( ( x  =  y  /\  x  e.  A )  ->  y  e.  A )
52, 4jca 519 . . . 4  |-  ( ( x  =  y  /\  x  e.  A )  ->  ( x  e.  A  /\  y  e.  A
) )
6 vex 2895 . . . . . 6  |-  y  e. 
_V
76opelres 5084 . . . . 5  |-  ( <.
x ,  y >.  e.  (  _I  |`  A )  <-> 
( <. x ,  y
>.  e.  _I  /\  x  e.  A ) )
8 df-br 4147 . . . . . . 7  |-  ( x  _I  y  <->  <. x ,  y >.  e.  _I  )
96ideq 4958 . . . . . . 7  |-  ( x  _I  y  <->  x  =  y )
108, 9bitr3i 243 . . . . . 6  |-  ( <.
x ,  y >.  e.  _I  <->  x  =  y
)
1110anbi1i 677 . . . . 5  |-  ( (
<. x ,  y >.  e.  _I  /\  x  e.  A )  <->  ( x  =  y  /\  x  e.  A ) )
127, 11bitri 241 . . . 4  |-  ( <.
x ,  y >.  e.  (  _I  |`  A )  <-> 
( x  =  y  /\  x  e.  A
) )
13 opelxp 4841 . . . 4  |-  ( <.
x ,  y >.  e.  ( A  X.  A
)  <->  ( x  e.  A  /\  y  e.  A ) )
145, 12, 133imtr4i 258 . . 3  |-  ( <.
x ,  y >.  e.  (  _I  |`  A )  ->  <. x ,  y
>.  e.  ( A  X.  A ) )
151, 14relssi 4900 . 2  |-  (  _I  |`  A )  C_  ( A  X.  A )
16 xpexg 4922 . . 3  |-  ( ( A  e.  V  /\  A  e.  V )  ->  ( A  X.  A
)  e.  _V )
1716anidms 627 . 2  |-  ( A  e.  V  ->  ( A  X.  A )  e. 
_V )
18 ssexg 4283 . 2  |-  ( ( (  _I  |`  A ) 
C_  ( A  X.  A )  /\  ( A  X.  A )  e. 
_V )  ->  (  _I  |`  A )  e. 
_V )
1915, 17, 18sylancr 645 1  |-  ( A  e.  V  ->  (  _I  |`  A )  e. 
_V )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    e. wcel 1717   _Vcvv 2892    C_ wss 3256   <.cop 3753   class class class wbr 4146    _I cid 4427    X. cxp 4809    |` cres 4813
This theorem is referenced by:  ordiso  7411  wdomref  7466  dfac9  7942  ndxarg  13409  idfu2nd  13994  idfu1st  13996  idfucl  13998  setcid  14161  pf1ind  19835  ausisusgra  21240  cusgraexilem1  21334  sizeusglecusg  21354  relexp0  24901  relexpsucr  24902  eldioph2lem1  26502  eldioph2lem2  26503  islinds2  26945  dib0  31330  dicn0  31358  cdlemn11a  31373  dihord6apre  31422  dihatlat  31500  dihpN  31502
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337  ax-un 4634
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-ral 2647  df-rex 2648  df-rab 2651  df-v 2894  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-op 3759  df-uni 3951  df-br 4147  df-opab 4201  df-id 4432  df-xp 4817  df-rel 4818  df-res 4823
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