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Theorem resfunexg 5671
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 5231 . . . . . . 7  |-  ( Fun 
A  ->  Fun  ( A  |`  B ) )
21adantr 453 . . . . . 6  |-  ( ( Fun  A  /\  B  e.  C )  ->  Fun  ( A  |`  B ) )
3 funfn 5222 . . . . . 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 5502 . . . . 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 5472 . . . . 5  |-  ( ( A  |`  B ) `  x )  e.  _V
87fnasrn 5636 . . . 4  |-  ( x  e.  dom  (  A  |`  B )  |->  ( ( A  |`  B ) `  x ) )  =  ran  (  x  e. 
dom  (  A  |`  B )  |->  <. x ,  ( ( A  |`  B ) `  x
) >. )
96, 8syl6eq 2306 . . 3  |-  ( ( Fun  A  /\  B  e.  C )  ->  ( A  |`  B )  =  ran  (  x  e. 
dom  (  A  |`  B )  |->  <. x ,  ( ( A  |`  B ) `  x
) >. ) )
10 opex 4209 . . . . . 6  |-  <. x ,  ( ( A  |`  B ) `  x
) >.  e.  _V
11 eqid 2258 . . . . . 6  |-  ( x  e.  dom  (  A  |`  B )  |->  <. x ,  ( ( A  |`  B ) `  x
) >. )  =  ( x  e.  dom  (  A  |`  B )  |->  <.
x ,  ( ( A  |`  B ) `  x ) >. )
1210, 11dmmpti 5311 . . . . 5  |-  dom  (  x  e.  dom  (  A  |`  B )  |->  <. x ,  ( ( A  |`  B ) `  x
) >. )  =  dom  (  A  |`  B )
1312imaeq2i 4998 . . . 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 5012 . . . 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 2280 . . 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 2308 . 2  |-  ( ( Fun  A  /\  B  e.  C )  ->  ( A  |`  B )  =  ( ( x  e. 
dom  (  A  |`  B )  |->  <. x ,  ( ( A  |`  B ) `  x
) >. ) " dom  (  A  |`  B ) ) )
17 funmpt 5229 . . 3  |-  Fun  (
x  e.  dom  (  A  |`  B )  |->  <.
x ,  ( ( A  |`  B ) `  x ) >. )
18 dmresexg 4966 . . . 4  |-  ( B  e.  C  ->  dom  (  A  |`  B )  e.  _V )
1918adantl 454 . . 3  |-  ( ( Fun  A  /\  B  e.  C )  ->  dom  (  A  |`  B )  e.  _V )
20 funimaexg 5267 . . 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 2332 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 2763   <.cop 3617    e. cmpt 4051   dom cdm 4661   ran crn 4662    |` cres 4663   "cima 4664   Fun wfun 4667    Fn wfn 4668   ` cfv 4673
This theorem is referenced by:  cofunexg  5673  fnex  5675  ofexg  6016  dfac8alem  7624  dfac12lem1  7737  cfsmolem  7864  alephsing  7870  itunifval  8010  zorn2lem1  8091  ttukeylem3  8106  imadomg  8127  wunex2  8328  inar1  8365  axdc4uzlem  11010  1stf1  13928  1stf2  13929  2ndf1  13931  2ndf2  13932  1stfcl  13933  2ndfcl  13934  bpolylem  24158  valtar  25250  idcatfun  25308  domidmor  25315  codidmor  25317  grphidmor  25319  dnnumch1  26508  aomclem6  26523  tendo02  30143
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 1927  ax-ext 2239  ax-rep 4105  ax-sep 4115  ax-nul 4123  ax-pr 4186  ax-un 4484
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 1884  df-eu 2122  df-mo 2123  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-ral 2523  df-rex 2524  df-reu 2525  df-rab 2527  df-v 2765  df-sbc 2967  df-csb 3057  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-nul 3431  df-if 3540  df-sn 3620  df-pr 3621  df-op 3623  df-uni 3802  df-iun 3881  df-br 3998  df-opab 4052  df-mpt 4053  df-id 4281  df-xp 4675  df-rel 4676  df-cnv 4677  df-co 4678  df-dm 4679  df-rn 4680  df-res 4681  df-ima 4682  df-fun 4683  df-fn 4684  df-f 4685  df-f1 4686  df-fo 4687  df-f1o 4688  df-fv 4689
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