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Theorem resfunexg 5589
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
StepHypRef Expression
1 funres 5150 . . . . . . 7  |-  ( Fun 
A  ->  Fun  ( A  |`  B ) )
21adantr 453 . . . . . 6  |-  ( ( Fun  A  /\  B  e.  C )  ->  Fun  ( A  |`  B ) )
3 funfn 5141 . . . . . 6  |-  ( Fun  ( A  |`  B )  <-> 
( A  |`  B )  Fn  dom  (  A  |`  B ) )
42, 3sylib 190 . . . . 5  |-  ( ( Fun  A  /\  B  e.  C )  ->  ( A  |`  B )  Fn 
dom  (  A  |`  B ) )
5 dffn5 5420 . . . . 5  |-  ( ( A  |`  B )  Fn  dom  (  A  |`  B )  <->  ( A  |`  B )  =  ( x  e.  dom  (  A  |`  B )  |->  ( ( A  |`  B ) `
 x ) ) )
64, 5sylib 190 . . . 4  |-  ( ( Fun  A  /\  B  e.  C )  ->  ( A  |`  B )  =  ( x  e.  dom  (  A  |`  B ) 
|->  ( ( A  |`  B ) `  x
) ) )
7 fvex 5391 . . . . 5  |-  ( ( A  |`  B ) `  x )  e.  _V
87fnasrn 5554 . . . 4  |-  ( x  e.  dom  (  A  |`  B )  |->  ( ( A  |`  B ) `  x ) )  =  ran  (  x  e. 
dom  (  A  |`  B )  |->  <. x ,  ( ( A  |`  B ) `  x
) >. )
96, 8syl6eq 2301 . . 3  |-  ( ( Fun  A  /\  B  e.  C )  ->  ( A  |`  B )  =  ran  (  x  e. 
dom  (  A  |`  B )  |->  <. x ,  ( ( A  |`  B ) `  x
) >. ) )
10 opex 4130 . . . . . 6  |-  <. x ,  ( ( A  |`  B ) `  x
) >.  e.  _V
11 eqid 2253 . . . . . 6  |-  ( x  e.  dom  (  A  |`  B )  |->  <. x ,  ( ( A  |`  B ) `  x
) >. )  =  ( x  e.  dom  (  A  |`  B )  |->  <.
x ,  ( ( A  |`  B ) `  x ) >. )
1210, 11dmmpti 5230 . . . . 5  |-  dom  (  x  e.  dom  (  A  |`  B )  |->  <. x ,  ( ( A  |`  B ) `  x
) >. )  =  dom  (  A  |`  B )
1312imaeq2i 4917 . . . 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 4931 . . . 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 2275 . . 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 2303 . 2  |-  ( ( Fun  A  /\  B  e.  C )  ->  ( A  |`  B )  =  ( ( x  e. 
dom  (  A  |`  B )  |->  <. x ,  ( ( A  |`  B ) `  x
) >. ) " dom  (  A  |`  B ) ) )
17 funmpt 5148 . . 3  |-  Fun  (
x  e.  dom  (  A  |`  B )  |->  <.
x ,  ( ( A  |`  B ) `  x ) >. )
18 dmresexg 4885 . . . 4  |-  ( B  e.  C  ->  dom  (  A  |`  B )  e.  _V )
1918adantl 454 . . 3  |-  ( ( Fun  A  /\  B  e.  C )  ->  dom  (  A  |`  B )  e.  _V )
20 funimaexg 5186 . . 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 647 . 2  |-  ( ( Fun  A  /\  B  e.  C )  ->  (
( x  e.  dom  (  A  |`  B ) 
|->  <. x ,  ( ( A  |`  B ) `
 x ) >.
) " dom  (  A  |`  B ) )  e.  _V )
2216, 21eqeltrd 2327 1  |-  ( ( Fun  A  /\  B  e.  C )  ->  ( A  |`  B )  e. 
_V )
Colors of variables: wff set class
Syntax hints:    -> wi 6    /\ wa 360    = wceq 1619    e. wcel 1621   _Vcvv 2727   <.cop 3547    e. cmpt 3974   dom cdm 4580   ran crn 4581    |` cres 4582   "cima 4583   Fun wfun 4586    Fn wfn 4587   ` cfv 4592
This theorem is referenced by:  cofunexg  5591  fnex  5593  ofexg  5934  dfac8alem  7540  dfac12lem1  7653  cfsmolem  7780  alephsing  7786  itunifval  7926  zorn2lem1  8007  ttukeylem3  8022  imadomg  8043  wunex2  8240  inar1  8277  axdc4uzlem  10922  1stf1  13810  1stf2  13811  2ndf1  13813  2ndf2  13814  1stfcl  13815  2ndfcl  13816  bpolylem  23957  valtar  25049  idcatfun  25107  domidmor  25114  codidmor  25116  grphidmor  25118  dnnumch1  26307  aomclem6  26322  tendo02  29880
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234  ax-rep 4028  ax-sep 4038  ax-nul 4046  ax-pr 4108  ax-un 4403
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2516  df-v 2729  df-sbc 2922  df-csb 3010  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-nul 3363  df-if 3471  df-sn 3550  df-pr 3551  df-op 3553  df-uni 3728  df-iun 3805  df-br 3921  df-opab 3975  df-mpt 3976  df-id 4202  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fn 4603  df-f 4604  df-f1 4605  df-fo 4606  df-f1o 4607  df-fv 4608
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