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Theorem pmresg 7458
Description: Elementhood of a restricted function in the set of partial functions. (Contributed by Mario Carneiro, 31-Dec-2013.)
Assertion
Ref Expression
pmresg  |-  ( ( B  e.  V  /\  F  e.  ( A  ^pm  C ) )  -> 
( F  |`  B )  e.  ( A  ^pm  B ) )

Proof of Theorem pmresg
StepHypRef Expression
1 n0i 3795 . . . . 5  |-  ( F  e.  ( A  ^pm  C )  ->  -.  ( A  ^pm  C )  =  (/) )
2 fnpm 7440 . . . . . . 7  |-  ^pm  Fn  ( _V  X.  _V )
3 fndm 5686 . . . . . . 7  |-  (  ^pm  Fn  ( _V  X.  _V )  ->  dom  ^pm  =  ( _V  X.  _V )
)
42, 3ax-mp 5 . . . . . 6  |-  dom  ^pm  =  ( _V  X.  _V )
54ndmov 6454 . . . . 5  |-  ( -.  ( A  e.  _V  /\  C  e.  _V )  ->  ( A  ^pm  C
)  =  (/) )
61, 5nsyl2 127 . . . 4  |-  ( F  e.  ( A  ^pm  C )  ->  ( A  e.  _V  /\  C  e. 
_V ) )
76simpld 459 . . 3  |-  ( F  e.  ( A  ^pm  C )  ->  A  e.  _V )
87adantl 466 . 2  |-  ( ( B  e.  V  /\  F  e.  ( A  ^pm  C ) )  ->  A  e.  _V )
9 simpl 457 . 2  |-  ( ( B  e.  V  /\  F  e.  ( A  ^pm  C ) )  ->  B  e.  V )
10 elpmi 7449 . . . . . 6  |-  ( F  e.  ( A  ^pm  C )  ->  ( F : dom  F --> A  /\  dom  F  C_  C )
)
1110simpld 459 . . . . 5  |-  ( F  e.  ( A  ^pm  C )  ->  F : dom  F --> A )
1211adantl 466 . . . 4  |-  ( ( B  e.  V  /\  F  e.  ( A  ^pm  C ) )  ->  F : dom  F --> A )
13 inss1 3723 . . . 4  |-  ( dom 
F  i^i  B )  C_ 
dom  F
14 fssres 5757 . . . 4  |-  ( ( F : dom  F --> A  /\  ( dom  F  i^i  B )  C_  dom  F )  ->  ( F  |`  ( dom  F  i^i  B ) ) : ( dom  F  i^i  B
) --> A )
1512, 13, 14sylancl 662 . . 3  |-  ( ( B  e.  V  /\  F  e.  ( A  ^pm  C ) )  -> 
( F  |`  ( dom  F  i^i  B ) ) : ( dom 
F  i^i  B ) --> A )
16 ffun 5739 . . . . 5  |-  ( F : dom  F --> A  ->  Fun  F )
17 resres 5292 . . . . . 6  |-  ( ( F  |`  dom  F )  |`  B )  =  ( F  |`  ( dom  F  i^i  B ) )
18 funrel 5611 . . . . . . 7  |-  ( Fun 
F  ->  Rel  F )
19 resdm 5321 . . . . . . 7  |-  ( Rel 
F  ->  ( F  |` 
dom  F )  =  F )
20 reseq1 5273 . . . . . . 7  |-  ( ( F  |`  dom  F )  =  F  ->  (
( F  |`  dom  F
)  |`  B )  =  ( F  |`  B ) )
2118, 19, 203syl 20 . . . . . 6  |-  ( Fun 
F  ->  ( ( F  |`  dom  F )  |`  B )  =  ( F  |`  B )
)
2217, 21syl5eqr 2522 . . . . 5  |-  ( Fun 
F  ->  ( F  |`  ( dom  F  i^i  B ) )  =  ( F  |`  B )
)
2312, 16, 223syl 20 . . . 4  |-  ( ( B  e.  V  /\  F  e.  ( A  ^pm  C ) )  -> 
( F  |`  ( dom  F  i^i  B ) )  =  ( F  |`  B ) )
2423feq1d 5723 . . 3  |-  ( ( B  e.  V  /\  F  e.  ( A  ^pm  C ) )  -> 
( ( F  |`  ( dom  F  i^i  B
) ) : ( dom  F  i^i  B
) --> A  <->  ( F  |`  B ) : ( dom  F  i^i  B
) --> A ) )
2515, 24mpbid 210 . 2  |-  ( ( B  e.  V  /\  F  e.  ( A  ^pm  C ) )  -> 
( F  |`  B ) : ( dom  F  i^i  B ) --> A )
26 inss2 3724 . . 3  |-  ( dom 
F  i^i  B )  C_  B
27 elpm2r 7448 . . 3  |-  ( ( ( A  e.  _V  /\  B  e.  V )  /\  ( ( F  |`  B ) : ( dom  F  i^i  B
) --> A  /\  ( dom  F  i^i  B ) 
C_  B ) )  ->  ( F  |`  B )  e.  ( A  ^pm  B )
)
2826, 27mpanr2 684 . 2  |-  ( ( ( A  e.  _V  /\  B  e.  V )  /\  ( F  |`  B ) : ( dom  F  i^i  B
) --> A )  -> 
( F  |`  B )  e.  ( A  ^pm  B ) )
298, 9, 25, 28syl21anc 1227 1  |-  ( ( B  e.  V  /\  F  e.  ( A  ^pm  C ) )  -> 
( F  |`  B )  e.  ( A  ^pm  B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767   _Vcvv 3118    i^i cin 3480    C_ wss 3481   (/)c0 3790    X. cxp 5003   dom cdm 5005    |` cres 5007   Rel wrel 5010   Fun wfun 5588    Fn wfn 5589   -->wf 5590  (class class class)co 6295    ^pm cpm 7433
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-fv 5602  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-1st 6795  df-2nd 6796  df-pm 7435
This theorem is referenced by:  lmres  19667  mbfres  21917  dvnres  22200  cpnres  22206  caures  30187
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