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Theorem resfunexg 5500
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 5041 . . . . 5  |-  ( Fun 
A  ->  Fun  ( A  |`  B ) )
2 funfvex 5306 . . . . . 6  |-  ( ( Fun  ( A  |`  B )  /\  x  e.  dom  ( A  |`  B ) )  -> 
( ( A  |`  B ) `  x
)  e.  _V )
32ralrimiva 2446 . . . . 5  |-  ( Fun  ( A  |`  B )  ->  A. x  e.  dom  ( A  |`  B ) ( ( A  |`  B ) `  x
)  e.  _V )
4 fnasrng 5461 . . . . 5  |-  ( A. x  e.  dom  ( A  |`  B ) ( ( A  |`  B ) `  x )  e.  _V  ->  ( x  e.  dom  ( A  |`  B ) 
|->  ( ( A  |`  B ) `  x
) )  =  ran  ( x  e.  dom  ( A  |`  B ) 
|->  <. x ,  ( ( A  |`  B ) `
 x ) >.
) )
51, 3, 43syl 17 . . . 4  |-  ( Fun 
A  ->  ( x  e.  dom  ( A  |`  B )  |->  ( ( A  |`  B ) `  x ) )  =  ran  ( x  e. 
dom  ( A  |`  B )  |->  <. x ,  ( ( A  |`  B ) `  x
) >. ) )
65adantr 270 . . 3  |-  ( ( Fun  A  /\  B  e.  C )  ->  (
x  e.  dom  ( A  |`  B )  |->  ( ( A  |`  B ) `
 x ) )  =  ran  ( x  e.  dom  ( A  |`  B )  |->  <. x ,  ( ( A  |`  B ) `  x
) >. ) )
71adantr 270 . . . . 5  |-  ( ( Fun  A  /\  B  e.  C )  ->  Fun  ( A  |`  B ) )
8 funfn 5031 . . . . 5  |-  ( Fun  ( A  |`  B )  <-> 
( A  |`  B )  Fn  dom  ( A  |`  B ) )
97, 8sylib 120 . . . 4  |-  ( ( Fun  A  /\  B  e.  C )  ->  ( A  |`  B )  Fn 
dom  ( A  |`  B ) )
10 dffn5im 5334 . . . 4  |-  ( ( A  |`  B )  Fn  dom  ( A  |`  B )  ->  ( A  |`  B )  =  ( x  e.  dom  ( A  |`  B ) 
|->  ( ( A  |`  B ) `  x
) ) )
119, 10syl 14 . . 3  |-  ( ( Fun  A  /\  B  e.  C )  ->  ( A  |`  B )  =  ( x  e.  dom  ( A  |`  B ) 
|->  ( ( A  |`  B ) `  x
) ) )
12 imadmrn 4771 . . . . 5  |-  ( ( 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
) >. )
13 vex 2622 . . . . . . . . 9  |-  x  e. 
_V
14 opexg 4046 . . . . . . . . 9  |-  ( ( x  e.  _V  /\  ( ( A  |`  B ) `  x
)  e.  _V )  -> 
<. x ,  ( ( A  |`  B ) `  x ) >.  e.  _V )
1513, 2, 14sylancr 405 . . . . . . . 8  |-  ( ( Fun  ( A  |`  B )  /\  x  e.  dom  ( A  |`  B ) )  ->  <. x ,  ( ( A  |`  B ) `  x ) >.  e.  _V )
1615ralrimiva 2446 . . . . . . 7  |-  ( Fun  ( A  |`  B )  ->  A. x  e.  dom  ( A  |`  B )
<. x ,  ( ( A  |`  B ) `  x ) >.  e.  _V )
17 dmmptg 4915 . . . . . . 7  |-  ( A. x  e.  dom  ( A  |`  B ) <. x ,  ( ( A  |`  B ) `  x
) >.  e.  _V  ->  dom  ( x  e.  dom  ( A  |`  B ) 
|->  <. x ,  ( ( A  |`  B ) `
 x ) >.
)  =  dom  ( A  |`  B ) )
181, 16, 173syl 17 . . . . . 6  |-  ( Fun 
A  ->  dom  ( x  e.  dom  ( A  |`  B )  |->  <. x ,  ( ( A  |`  B ) `  x
) >. )  =  dom  ( A  |`  B ) )
1918imaeq2d 4761 . . . . 5  |-  ( Fun 
A  ->  ( (
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 ) ) )
2012, 19syl5reqr 2135 . . . 4  |-  ( Fun 
A  ->  ( (
x  e.  dom  ( A  |`  B )  |->  <.
x ,  ( ( A  |`  B ) `  x ) >. ) " dom  ( A  |`  B ) )  =  ran  ( x  e. 
dom  ( A  |`  B )  |->  <. x ,  ( ( A  |`  B ) `  x
) >. ) )
2120adantr 270 . . 3  |-  ( ( Fun  A  /\  B  e.  C )  ->  (
( x  e.  dom  ( A  |`  B ) 
|->  <. x ,  ( ( A  |`  B ) `
 x ) >.
) " dom  ( A  |`  B ) )  =  ran  ( x  e.  dom  ( A  |`  B )  |->  <. x ,  ( ( A  |`  B ) `  x
) >. ) )
226, 11, 213eqtr4d 2130 . 2  |-  ( ( Fun  A  /\  B  e.  C )  ->  ( A  |`  B )  =  ( ( x  e. 
dom  ( A  |`  B )  |->  <. x ,  ( ( A  |`  B ) `  x
) >. ) " dom  ( A  |`  B ) ) )
23 funmpt 5038 . . 3  |-  Fun  (
x  e.  dom  ( A  |`  B )  |->  <.
x ,  ( ( A  |`  B ) `  x ) >. )
24 dmresexg 4723 . . . 4  |-  ( B  e.  C  ->  dom  ( A  |`  B )  e.  _V )
2524adantl 271 . . 3  |-  ( ( Fun  A  /\  B  e.  C )  ->  dom  ( A  |`  B )  e.  _V )
26 funimaexg 5084 . . 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 )
2723, 25, 26sylancr 405 . 2  |-  ( ( Fun  A  /\  B  e.  C )  ->  (
( x  e.  dom  ( A  |`  B ) 
|->  <. x ,  ( ( A  |`  B ) `
 x ) >.
) " dom  ( A  |`  B ) )  e.  _V )
2822, 27eqeltrd 2164 1  |-  ( ( Fun  A  /\  B  e.  C )  ->  ( A  |`  B )  e. 
_V )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    = wceq 1289    e. wcel 1438   A.wral 2359   _Vcvv 2619   <.cop 3444    |-> cmpt 3891   dom cdm 4428   ran crn 4429    |` cres 4430   "cima 4431   Fun wfun 4996    Fn wfn 4997   ` cfv 5002
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-coll 3946  ax-sep 3949  ax-pow 4001  ax-pr 4027
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-reu 2366  df-rab 2368  df-v 2621  df-sbc 2839  df-csb 2932  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-iun 3727  df-br 3838  df-opab 3892  df-mpt 3893  df-id 4111  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-rn 4439  df-res 4440  df-ima 4441  df-iota 4967  df-fun 5004  df-fn 5005  df-f 5006  df-f1 5007  df-fo 5008  df-f1o 5009  df-fv 5010
This theorem is referenced by:  fnex  5501  ofexg  5842  cofunexg  5864  rdgivallem  6128  frecex  6141  frecsuclem  6153  fihashf1rn  10162
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