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Theorem resfunexg 5916
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 5451 . . . . . . 7  |-  ( Fun 
A  ->  Fun  ( A  |`  B ) )
21adantr 452 . . . . . 6  |-  ( ( Fun  A  /\  B  e.  C )  ->  Fun  ( A  |`  B ) )
3 funfn 5441 . . . . . 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 5731 . . . . 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 5701 . . . . 5  |-  ( ( A  |`  B ) `  x )  e.  _V
87fnasrn 5871 . . . 4  |-  ( x  e.  dom  ( A  |`  B )  |->  ( ( A  |`  B ) `  x ) )  =  ran  ( x  e. 
dom  ( A  |`  B )  |->  <. x ,  ( ( A  |`  B ) `  x
) >. )
96, 8syl6eq 2452 . . 3  |-  ( ( Fun  A  /\  B  e.  C )  ->  ( A  |`  B )  =  ran  ( x  e. 
dom  ( A  |`  B )  |->  <. x ,  ( ( A  |`  B ) `  x
) >. ) )
10 opex 4387 . . . . . 6  |-  <. x ,  ( ( A  |`  B ) `  x
) >.  e.  _V
11 eqid 2404 . . . . . 6  |-  ( x  e.  dom  ( A  |`  B )  |->  <. x ,  ( ( A  |`  B ) `  x
) >. )  =  ( x  e.  dom  ( A  |`  B )  |->  <.
x ,  ( ( A  |`  B ) `  x ) >. )
1210, 11dmmpti 5533 . . . . 5  |-  dom  (
x  e.  dom  ( A  |`  B )  |->  <.
x ,  ( ( A  |`  B ) `  x ) >. )  =  dom  ( A  |`  B )
1312imaeq2i 5160 . . . 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 5174 . . . 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 2426 . . 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 2454 . 2  |-  ( ( Fun  A  /\  B  e.  C )  ->  ( A  |`  B )  =  ( ( x  e. 
dom  ( A  |`  B )  |->  <. x ,  ( ( A  |`  B ) `  x
) >. ) " dom  ( A  |`  B ) ) )
17 funmpt 5448 . . 3  |-  Fun  (
x  e.  dom  ( A  |`  B )  |->  <.
x ,  ( ( A  |`  B ) `  x ) >. )
18 dmresexg 5128 . . . 4  |-  ( B  e.  C  ->  dom  ( A  |`  B )  e.  _V )
1918adantl 453 . . 3  |-  ( ( Fun  A  /\  B  e.  C )  ->  dom  ( A  |`  B )  e.  _V )
20 funimaexg 5489 . . 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 2478 1  |-  ( ( Fun  A  /\  B  e.  C )  ->  ( A  |`  B )  e. 
_V )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1721   _Vcvv 2916   <.cop 3777    e. cmpt 4226   dom cdm 4837   ran crn 4838    |` cres 4839   "cima 4840   Fun wfun 5407    Fn wfn 5408   ` cfv 5413
This theorem is referenced by:  cofunexg  5918  fnex  5920  ofexg  6268  dfac8alem  7866  dfac12lem1  7979  cfsmolem  8106  alephsing  8112  itunifval  8252  zorn2lem1  8332  ttukeylem3  8347  imadomg  8368  wunex2  8569  inar1  8606  axdc4uzlem  11276  hashf1rn  11591  1stf1  14244  1stf2  14245  2ndf1  14247  2ndf2  14248  1stfcl  14249  2ndfcl  14250  bpolylem  25998  dnnumch1  27009  aomclem6  27024  tendo02  31269
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pr 4363
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421
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