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Theorem resfunexg 6145
Description: The restriction of a function to a set exists. Compare Proposition 6.17 of [TakeutiZaring] p. 28. (Contributed by NM, 7-Apr-1995.) (Revised by Mario Carneiro, 22-Jun-2013.)
Assertion
Ref Expression
resfunexg  |-  ( ( Fun  A  /\  B  e.  C )  ->  ( A  |`  B )  e. 
_V )

Proof of Theorem resfunexg
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 funres 5640 . . . . . . 7  |-  ( Fun 
A  ->  Fun  ( A  |`  B ) )
21adantr 466 . . . . . 6  |-  ( ( Fun  A  /\  B  e.  C )  ->  Fun  ( A  |`  B ) )
3 funfn 5630 . . . . . 6  |-  ( Fun  ( A  |`  B )  <-> 
( A  |`  B )  Fn  dom  ( A  |`  B ) )
42, 3sylib 199 . . . . 5  |-  ( ( Fun  A  /\  B  e.  C )  ->  ( A  |`  B )  Fn 
dom  ( A  |`  B ) )
5 dffn5 5926 . . . . 5  |-  ( ( A  |`  B )  Fn  dom  ( A  |`  B )  <->  ( A  |`  B )  =  ( x  e.  dom  ( A  |`  B )  |->  ( ( A  |`  B ) `
 x ) ) )
64, 5sylib 199 . . . 4  |-  ( ( Fun  A  /\  B  e.  C )  ->  ( A  |`  B )  =  ( x  e.  dom  ( A  |`  B ) 
|->  ( ( A  |`  B ) `  x
) ) )
7 fvex 5891 . . . . 5  |-  ( ( A  |`  B ) `  x )  e.  _V
87fnasrn 6085 . . . 4  |-  ( x  e.  dom  ( A  |`  B )  |->  ( ( A  |`  B ) `  x ) )  =  ran  ( x  e. 
dom  ( A  |`  B )  |->  <. x ,  ( ( A  |`  B ) `  x
) >. )
96, 8syl6eq 2486 . . 3  |-  ( ( Fun  A  /\  B  e.  C )  ->  ( A  |`  B )  =  ran  ( x  e. 
dom  ( A  |`  B )  |->  <. x ,  ( ( A  |`  B ) `  x
) >. ) )
10 opex 4686 . . . . . 6  |-  <. x ,  ( ( A  |`  B ) `  x
) >.  e.  _V
11 eqid 2429 . . . . . 6  |-  ( x  e.  dom  ( A  |`  B )  |->  <. x ,  ( ( A  |`  B ) `  x
) >. )  =  ( x  e.  dom  ( A  |`  B )  |->  <.
x ,  ( ( A  |`  B ) `  x ) >. )
1210, 11dmmpti 5725 . . . . 5  |-  dom  (
x  e.  dom  ( A  |`  B )  |->  <.
x ,  ( ( A  |`  B ) `  x ) >. )  =  dom  ( A  |`  B )
1312imaeq2i 5186 . . . 4  |-  ( ( x  e.  dom  ( A  |`  B )  |->  <.
x ,  ( ( A  |`  B ) `  x ) >. ) " dom  ( x  e. 
dom  ( A  |`  B )  |->  <. x ,  ( ( A  |`  B ) `  x
) >. ) )  =  ( ( x  e. 
dom  ( A  |`  B )  |->  <. x ,  ( ( A  |`  B ) `  x
) >. ) " dom  ( A  |`  B ) )
14 imadmrn 5198 . . . 4  |-  ( ( x  e.  dom  ( A  |`  B )  |->  <.
x ,  ( ( A  |`  B ) `  x ) >. ) " dom  ( x  e. 
dom  ( A  |`  B )  |->  <. x ,  ( ( A  |`  B ) `  x
) >. ) )  =  ran  ( x  e. 
dom  ( A  |`  B )  |->  <. x ,  ( ( A  |`  B ) `  x
) >. )
1513, 14eqtr3i 2460 . . 3  |-  ( ( x  e.  dom  ( A  |`  B )  |->  <.
x ,  ( ( A  |`  B ) `  x ) >. ) " dom  ( A  |`  B ) )  =  ran  ( x  e. 
dom  ( A  |`  B )  |->  <. x ,  ( ( A  |`  B ) `  x
) >. )
169, 15syl6eqr 2488 . 2  |-  ( ( Fun  A  /\  B  e.  C )  ->  ( A  |`  B )  =  ( ( x  e. 
dom  ( A  |`  B )  |->  <. x ,  ( ( A  |`  B ) `  x
) >. ) " dom  ( A  |`  B ) ) )
17 funmpt 5637 . . 3  |-  Fun  (
x  e.  dom  ( A  |`  B )  |->  <.
x ,  ( ( A  |`  B ) `  x ) >. )
18 dmresexg 5147 . . . 4  |-  ( B  e.  C  ->  dom  ( A  |`  B )  e.  _V )
1918adantl 467 . . 3  |-  ( ( Fun  A  /\  B  e.  C )  ->  dom  ( A  |`  B )  e.  _V )
20 funimaexg 5678 . . 3  |-  ( ( Fun  ( x  e. 
dom  ( A  |`  B )  |->  <. x ,  ( ( A  |`  B ) `  x
) >. )  /\  dom  ( A  |`  B )  e.  _V )  -> 
( ( x  e. 
dom  ( A  |`  B )  |->  <. x ,  ( ( A  |`  B ) `  x
) >. ) " dom  ( A  |`  B ) )  e.  _V )
2117, 19, 20sylancr 667 . 2  |-  ( ( Fun  A  /\  B  e.  C )  ->  (
( x  e.  dom  ( A  |`  B ) 
|->  <. x ,  ( ( A  |`  B ) `
 x ) >.
) " dom  ( A  |`  B ) )  e.  _V )
2216, 21eqeltrd 2517 1  |-  ( ( Fun  A  /\  B  e.  C )  ->  ( A  |`  B )  e. 
_V )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    = wceq 1437    e. wcel 1870   _Vcvv 3087   <.cop 4008    |-> cmpt 4484   dom cdm 4854   ran crn 4855    |` cres 4856   "cima 4857   Fun wfun 5595    Fn wfn 5596   ` cfv 5601
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pr 4661
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-reu 2789  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-id 4769  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609
This theorem is referenced by:  resiexd  6146  fnex  6147  ofexg  6549  cofunexg  6771  dfac8alem  8458  dfac12lem1  8571  cfsmolem  8698  alephsing  8704  itunifval  8844  zorn2lem1  8924  ttukeylem3  8939  imadomg  8960  wunex2  9162  inar1  9199  axdc4uzlem  12192  hashf1rn  12532  bpolylem  14079  1stf1  16028  1stf2  16029  2ndf1  16031  2ndf2  16032  1stfcl  16033  2ndfcl  16034  gsumzadd  17490  tendo02  34063  dnnumch1  35608  aomclem6  35623  usgresvm1  38513  usgresvm1ALT  38517  dfrngc2  38732  dfringc2  38778  rngcresringcat  38790  fdivval  39111
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