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Theorem resfunexg 5408
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 4965 . . . . 5  |-  ( Fun 
A  ->  Fun  ( A  |`  B ) )
2 funfvex 5217 . . . . . 6  |-  ( ( Fun  ( A  |`  B )  /\  x  e.  dom  ( A  |`  B ) )  -> 
( ( A  |`  B ) `  x
)  e.  _V )
32ralrimiva 2435 . . . . 5  |-  ( Fun  ( A  |`  B )  ->  A. x  e.  dom  ( A  |`  B ) ( ( A  |`  B ) `  x
)  e.  _V )
4 fnasrng 5369 . . . . 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 4955 . . . . 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 5245 . . . 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 4702 . . . . 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 2605 . . . . . . . . 9  |-  x  e. 
_V
14 opexg 3985 . . . . . . . . 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 2435 . . . . . . 7  |-  ( Fun  ( A  |`  B )  ->  A. x  e.  dom  ( A  |`  B )
<. x ,  ( ( A  |`  B ) `  x ) >.  e.  _V )
17 dmmptg 4842 . . . . . . 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 4692 . . . . 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 2129 . . . 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 2124 . 2  |-  ( ( Fun  A  /\  B  e.  C )  ->  ( A  |`  B )  =  ( ( x  e. 
dom  ( A  |`  B )  |->  <. x ,  ( ( A  |`  B ) `  x
) >. ) " dom  ( A  |`  B ) ) )
23 funmpt 4962 . . 3  |-  Fun  (
x  e.  dom  ( A  |`  B )  |->  <.
x ,  ( ( A  |`  B ) `  x ) >. )
24 dmresexg 4656 . . . 4  |-  ( B  e.  C  ->  dom  ( A  |`  B )  e.  _V )
2524adantl 271 . . 3  |-  ( ( Fun  A  /\  B  e.  C )  ->  dom  ( A  |`  B )  e.  _V )
26 funimaexg 5008 . . 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 2156 1  |-  ( ( Fun  A  /\  B  e.  C )  ->  ( A  |`  B )  e. 
_V )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    = wceq 1285    e. wcel 1434   A.wral 2349   _Vcvv 2602   <.cop 3403    |-> cmpt 3841   dom cdm 4365   ran crn 4366    |` cres 4367   "cima 4368   Fun wfun 4920    Fn wfn 4921   ` cfv 4926
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 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-coll 3895  ax-sep 3898  ax-pow 3950  ax-pr 3966
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rex 2355  df-reu 2356  df-rab 2358  df-v 2604  df-sbc 2817  df-csb 2910  df-un 2978  df-in 2980  df-ss 2987  df-pw 3386  df-sn 3406  df-pr 3407  df-op 3409  df-uni 3604  df-iun 3682  df-br 3788  df-opab 3842  df-mpt 3843  df-id 4050  df-xp 4371  df-rel 4372  df-cnv 4373  df-co 4374  df-dm 4375  df-rn 4376  df-res 4377  df-ima 4378  df-iota 4891  df-fun 4928  df-fn 4929  df-f 4930  df-f1 4931  df-fo 4932  df-f1o 4933  df-fv 4934
This theorem is referenced by:  fnex  5409  ofexg  5741  cofunexg  5763  rdgivallem  6024  frecex  6037  frecsuclem  6049  sizef1rn  9802
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