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Theorem resiexg 6748
Description: The existence of a restricted identity function, proved without using the Axiom of Replacement (unlike resfunexg 6146). (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 5138 . . 3  |-  Rel  (  _I  |`  A )
2 simpr 468 . . . . 5  |-  ( ( x  =  y  /\  x  e.  A )  ->  x  e.  A )
3 eleq1 2537 . . . . . 6  |-  ( x  =  y  ->  (
x  e.  A  <->  y  e.  A ) )
43biimpa 492 . . . . 5  |-  ( ( x  =  y  /\  x  e.  A )  ->  y  e.  A )
52, 4jca 541 . . . 4  |-  ( ( x  =  y  /\  x  e.  A )  ->  ( x  e.  A  /\  y  e.  A
) )
6 vex 3034 . . . . . 6  |-  y  e. 
_V
76opelres 5116 . . . . 5  |-  ( <.
x ,  y >.  e.  (  _I  |`  A )  <-> 
( <. x ,  y
>.  e.  _I  /\  x  e.  A ) )
8 df-br 4396 . . . . . . 7  |-  ( x  _I  y  <->  <. x ,  y >.  e.  _I  )
96ideq 4992 . . . . . . 7  |-  ( x  _I  y  <->  x  =  y )
108, 9bitr3i 259 . . . . . 6  |-  ( <.
x ,  y >.  e.  _I  <->  x  =  y
)
1110anbi1i 709 . . . . 5  |-  ( (
<. x ,  y >.  e.  _I  /\  x  e.  A )  <->  ( x  =  y  /\  x  e.  A ) )
127, 11bitri 257 . . . 4  |-  ( <.
x ,  y >.  e.  (  _I  |`  A )  <-> 
( x  =  y  /\  x  e.  A
) )
13 opelxp 4869 . . . 4  |-  ( <.
x ,  y >.  e.  ( A  X.  A
)  <->  ( x  e.  A  /\  y  e.  A ) )
145, 12, 133imtr4i 274 . . 3  |-  ( <.
x ,  y >.  e.  (  _I  |`  A )  ->  <. x ,  y
>.  e.  ( A  X.  A ) )
151, 14relssi 4931 . 2  |-  (  _I  |`  A )  C_  ( A  X.  A )
16 sqxpexg 6615 . 2  |-  ( A  e.  V  ->  ( A  X.  A )  e. 
_V )
17 ssexg 4542 . 2  |-  ( ( (  _I  |`  A ) 
C_  ( A  X.  A )  /\  ( A  X.  A )  e. 
_V )  ->  (  _I  |`  A )  e. 
_V )
1815, 16, 17sylancr 676 1  |-  ( A  e.  V  ->  (  _I  |`  A )  e. 
_V )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 376    e. wcel 1904   _Vcvv 3031    C_ wss 3390   <.cop 3965   class class class wbr 4395    _I cid 4749    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 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-br 4396  df-opab 4455  df-id 4754  df-xp 4845  df-rel 4846  df-res 4851
This theorem is referenced by:  ordiso  8049  wdomref  8105  dfac9  8584  relexp0g  13162  relexpsucnnr  13165  ndxarg  15219  idfu2nd  15860  idfu1st  15862  idfucl  15864  setcid  16059  equivestrcsetc  16115  pf1ind  19020  islinds2  19448  ausisusgra  25161  cusgraexilem1  25273  sizeusglecusg  25293  poimirlem15  32019  dib0  34803  dicn0  34831  cdlemn11a  34846  dihord6apre  34895  dihatlat  34973  dihpN  34975  eldioph2lem1  35673  eldioph2lem2  35674  dfrtrcl5  36307  dfrcl2  36337  relexpiidm  36367  ausgrusgrb  39413  upgrres1lem1  39540  usgrexi  39671  sizusglecusg  39689  rngcidALTV  40501  ringcidALTV  40564
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