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Theorem resfunexg 5957
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 5492 . . . . . . 7  |-  ( Fun 
A  ->  Fun  ( A  |`  B ) )
21adantr 452 . . . . . 6  |-  ( ( Fun  A  /\  B  e.  C )  ->  Fun  ( A  |`  B ) )
3 funfn 5482 . . . . . 6  |-  ( Fun  ( A  |`  B )  <-> 
( A  |`  B )  Fn  dom  ( A  |`  B ) )
42, 3sylib 189 . . . . 5  |-  ( ( Fun  A  /\  B  e.  C )  ->  ( A  |`  B )  Fn 
dom  ( A  |`  B ) )
5 dffn5 5772 . . . . 5  |-  ( ( A  |`  B )  Fn  dom  ( A  |`  B )  <->  ( A  |`  B )  =  ( x  e.  dom  ( A  |`  B )  |->  ( ( A  |`  B ) `
 x ) ) )
64, 5sylib 189 . . . 4  |-  ( ( Fun  A  /\  B  e.  C )  ->  ( A  |`  B )  =  ( x  e.  dom  ( A  |`  B ) 
|->  ( ( A  |`  B ) `  x
) ) )
7 fvex 5742 . . . . 5  |-  ( ( A  |`  B ) `  x )  e.  _V
87fnasrn 5912 . . . 4  |-  ( x  e.  dom  ( A  |`  B )  |->  ( ( A  |`  B ) `  x ) )  =  ran  ( x  e. 
dom  ( A  |`  B )  |->  <. x ,  ( ( A  |`  B ) `  x
) >. )
96, 8syl6eq 2484 . . 3  |-  ( ( Fun  A  /\  B  e.  C )  ->  ( A  |`  B )  =  ran  ( x  e. 
dom  ( A  |`  B )  |->  <. x ,  ( ( A  |`  B ) `  x
) >. ) )
10 opex 4427 . . . . . 6  |-  <. x ,  ( ( A  |`  B ) `  x
) >.  e.  _V
11 eqid 2436 . . . . . 6  |-  ( x  e.  dom  ( A  |`  B )  |->  <. x ,  ( ( A  |`  B ) `  x
) >. )  =  ( x  e.  dom  ( A  |`  B )  |->  <.
x ,  ( ( A  |`  B ) `  x ) >. )
1210, 11dmmpti 5574 . . . . 5  |-  dom  (
x  e.  dom  ( A  |`  B )  |->  <.
x ,  ( ( A  |`  B ) `  x ) >. )  =  dom  ( A  |`  B )
1312imaeq2i 5201 . . . 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 5215 . . . 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 2458 . . 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 2486 . 2  |-  ( ( Fun  A  /\  B  e.  C )  ->  ( A  |`  B )  =  ( ( x  e. 
dom  ( A  |`  B )  |->  <. x ,  ( ( A  |`  B ) `  x
) >. ) " dom  ( A  |`  B ) ) )
17 funmpt 5489 . . 3  |-  Fun  (
x  e.  dom  ( A  |`  B )  |->  <.
x ,  ( ( A  |`  B ) `  x ) >. )
18 dmresexg 5169 . . . 4  |-  ( B  e.  C  ->  dom  ( A  |`  B )  e.  _V )
1918adantl 453 . . 3  |-  ( ( Fun  A  /\  B  e.  C )  ->  dom  ( A  |`  B )  e.  _V )
20 funimaexg 5530 . . 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 645 . 2  |-  ( ( Fun  A  /\  B  e.  C )  ->  (
( x  e.  dom  ( A  |`  B ) 
|->  <. x ,  ( ( A  |`  B ) `
 x ) >.
) " dom  ( A  |`  B ) )  e.  _V )
2216, 21eqeltrd 2510 1  |-  ( ( Fun  A  /\  B  e.  C )  ->  ( A  |`  B )  e. 
_V )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   _Vcvv 2956   <.cop 3817    e. cmpt 4266   dom cdm 4878   ran crn 4879    |` cres 4880   "cima 4881   Fun wfun 5448    Fn wfn 5449   ` cfv 5454
This theorem is referenced by:  cofunexg  5959  fnex  5961  ofexg  6309  dfac8alem  7910  dfac12lem1  8023  cfsmolem  8150  alephsing  8156  itunifval  8296  zorn2lem1  8376  ttukeylem3  8391  imadomg  8412  wunex2  8613  inar1  8650  axdc4uzlem  11321  hashf1rn  11636  1stf1  14289  1stf2  14290  2ndf1  14292  2ndf2  14293  1stfcl  14294  2ndfcl  14295  bpolylem  26094  dnnumch1  27119  aomclem6  27134  tendo02  31584
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pr 4403
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462
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