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Theorem resfunexg 5753
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 5272 . . . . 5  |-  ( Fun 
A  ->  Fun  ( A  |`  B ) )
2 funfvex 5547 . . . . . 6  |-  ( ( Fun  ( A  |`  B )  /\  x  e.  dom  ( A  |`  B ) )  -> 
( ( A  |`  B ) `  x
)  e.  _V )
32ralrimiva 2563 . . . . 5  |-  ( Fun  ( A  |`  B )  ->  A. x  e.  dom  ( A  |`  B ) ( ( A  |`  B ) `  x
)  e.  _V )
4 fnasrng 5712 . . . . 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 276 . . 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 276 . . . . 5  |-  ( ( Fun  A  /\  B  e.  C )  ->  Fun  ( A  |`  B ) )
8 funfn 5261 . . . . 5  |-  ( Fun  ( A  |`  B )  <-> 
( A  |`  B )  Fn  dom  ( A  |`  B ) )
97, 8sylib 122 . . . 4  |-  ( ( Fun  A  /\  B  e.  C )  ->  ( A  |`  B )  Fn 
dom  ( A  |`  B ) )
10 dffn5im 5577 . . . 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 vex 2755 . . . . . . . . 9  |-  x  e. 
_V
13 opexg 4243 . . . . . . . . 9  |-  ( ( x  e.  _V  /\  ( ( A  |`  B ) `  x
)  e.  _V )  -> 
<. x ,  ( ( A  |`  B ) `  x ) >.  e.  _V )
1412, 2, 13sylancr 414 . . . . . . . 8  |-  ( ( Fun  ( A  |`  B )  /\  x  e.  dom  ( A  |`  B ) )  ->  <. x ,  ( ( A  |`  B ) `  x ) >.  e.  _V )
1514ralrimiva 2563 . . . . . . 7  |-  ( Fun  ( A  |`  B )  ->  A. x  e.  dom  ( A  |`  B )
<. x ,  ( ( A  |`  B ) `  x ) >.  e.  _V )
16 dmmptg 5141 . . . . . . 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 ) )
171, 15, 163syl 17 . . . . . 6  |-  ( Fun 
A  ->  dom  ( x  e.  dom  ( A  |`  B )  |->  <. x ,  ( ( A  |`  B ) `  x
) >. )  =  dom  ( A  |`  B ) )
1817imaeq2d 4985 . . . . 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 ) ) )
19 imadmrn 4995 . . . . 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
) >. )
2018, 19eqtr3di 2237 . . . 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 276 . . 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 2232 . 2  |-  ( ( Fun  A  /\  B  e.  C )  ->  ( A  |`  B )  =  ( ( x  e. 
dom  ( A  |`  B )  |->  <. x ,  ( ( A  |`  B ) `  x
) >. ) " dom  ( A  |`  B ) ) )
23 funmpt 5269 . . 3  |-  Fun  (
x  e.  dom  ( A  |`  B )  |->  <.
x ,  ( ( A  |`  B ) `  x ) >. )
24 dmresexg 4945 . . . 4  |-  ( B  e.  C  ->  dom  ( A  |`  B )  e.  _V )
2524adantl 277 . . 3  |-  ( ( Fun  A  /\  B  e.  C )  ->  dom  ( A  |`  B )  e.  _V )
26 funimaexg 5315 . . 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 414 . 2  |-  ( ( Fun  A  /\  B  e.  C )  ->  (
( x  e.  dom  ( A  |`  B ) 
|->  <. x ,  ( ( A  |`  B ) `
 x ) >.
) " dom  ( A  |`  B ) )  e.  _V )
2822, 27eqeltrd 2266 1  |-  ( ( Fun  A  /\  B  e.  C )  ->  ( A  |`  B )  e. 
_V )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    = wceq 1364    e. wcel 2160   A.wral 2468   _Vcvv 2752   <.cop 3610    |-> cmpt 4079   dom cdm 4641   ran crn 4642    |` cres 4643   "cima 4644   Fun wfun 5225    Fn wfn 5226   ` cfv 5231
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-pow 4189  ax-pr 4224
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-reu 2475  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-id 4308  df-xp 4647  df-rel 4648  df-cnv 4649  df-co 4650  df-dm 4651  df-rn 4652  df-res 4653  df-ima 4654  df-iota 5193  df-fun 5233  df-fn 5234  df-f 5235  df-f1 5236  df-fo 5237  df-f1o 5238  df-fv 5239
This theorem is referenced by:  fnex  5754  ofexg  6105  cofunexg  6128  rdgivallem  6400  frecex  6413  frecsuclem  6425  djudoml  7237  djudomr  7238  fihashf1rn  10787  qnnen  12456
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