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Theorem resfunexg 5607
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 5132 . . . . 5  |-  ( Fun 
A  ->  Fun  ( A  |`  B ) )
2 funfvex 5404 . . . . . 6  |-  ( ( Fun  ( A  |`  B )  /\  x  e.  dom  ( A  |`  B ) )  -> 
( ( A  |`  B ) `  x
)  e.  _V )
32ralrimiva 2480 . . . . 5  |-  ( Fun  ( A  |`  B )  ->  A. x  e.  dom  ( A  |`  B ) ( ( A  |`  B ) `  x
)  e.  _V )
4 fnasrng 5566 . . . . 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 272 . . 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 272 . . . . 5  |-  ( ( Fun  A  /\  B  e.  C )  ->  Fun  ( A  |`  B ) )
8 funfn 5121 . . . . 5  |-  ( Fun  ( A  |`  B )  <-> 
( A  |`  B )  Fn  dom  ( A  |`  B ) )
97, 8sylib 121 . . . 4  |-  ( ( Fun  A  /\  B  e.  C )  ->  ( A  |`  B )  Fn 
dom  ( A  |`  B ) )
10 dffn5im 5433 . . . 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 4859 . . . . 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 2661 . . . . . . . . 9  |-  x  e. 
_V
14 opexg 4118 . . . . . . . . 9  |-  ( ( x  e.  _V  /\  ( ( A  |`  B ) `  x
)  e.  _V )  -> 
<. x ,  ( ( A  |`  B ) `  x ) >.  e.  _V )
1513, 2, 14sylancr 408 . . . . . . . 8  |-  ( ( Fun  ( A  |`  B )  /\  x  e.  dom  ( A  |`  B ) )  ->  <. x ,  ( ( A  |`  B ) `  x ) >.  e.  _V )
1615ralrimiva 2480 . . . . . . 7  |-  ( Fun  ( A  |`  B )  ->  A. x  e.  dom  ( A  |`  B )
<. x ,  ( ( A  |`  B ) `  x ) >.  e.  _V )
17 dmmptg 5004 . . . . . . 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 4849 . . . . 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 2163 . . . 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 272 . . 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 2158 . 2  |-  ( ( Fun  A  /\  B  e.  C )  ->  ( A  |`  B )  =  ( ( x  e. 
dom  ( A  |`  B )  |->  <. x ,  ( ( A  |`  B ) `  x
) >. ) " dom  ( A  |`  B ) ) )
23 funmpt 5129 . . 3  |-  Fun  (
x  e.  dom  ( A  |`  B )  |->  <.
x ,  ( ( A  |`  B ) `  x ) >. )
24 dmresexg 4810 . . . 4  |-  ( B  e.  C  ->  dom  ( A  |`  B )  e.  _V )
2524adantl 273 . . 3  |-  ( ( Fun  A  /\  B  e.  C )  ->  dom  ( A  |`  B )  e.  _V )
26 funimaexg 5175 . . 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 408 . 2  |-  ( ( Fun  A  /\  B  e.  C )  ->  (
( x  e.  dom  ( A  |`  B ) 
|->  <. x ,  ( ( A  |`  B ) `
 x ) >.
) " dom  ( A  |`  B ) )  e.  _V )
2822, 27eqeltrd 2192 1  |-  ( ( Fun  A  /\  B  e.  C )  ->  ( A  |`  B )  e. 
_V )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1314    e. wcel 1463   A.wral 2391   _Vcvv 2658   <.cop 3498    |-> cmpt 3957   dom cdm 4507   ran crn 4508    |` cres 4509   "cima 4510   Fun wfun 5085    Fn wfn 5086   ` cfv 5091
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-coll 4011  ax-sep 4014  ax-pow 4066  ax-pr 4099
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396  df-rex 2397  df-reu 2398  df-rab 2400  df-v 2660  df-sbc 2881  df-csb 2974  df-un 3043  df-in 3045  df-ss 3052  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-iun 3783  df-br 3898  df-opab 3958  df-mpt 3959  df-id 4183  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-rn 4518  df-res 4519  df-ima 4520  df-iota 5056  df-fun 5093  df-fn 5094  df-f 5095  df-f1 5096  df-fo 5097  df-f1o 5098  df-fv 5099
This theorem is referenced by:  fnex  5608  ofexg  5952  cofunexg  5975  rdgivallem  6244  frecex  6257  frecsuclem  6269  djudoml  7039  djudomr  7040  fihashf1rn  10486  qnnen  11850
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