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Theorem pmresg 7498
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 3763 . . . . 5  |-  ( F  e.  ( A  ^pm  C )  ->  -.  ( A  ^pm  C )  =  (/) )
2 fnpm 7479 . . . . . . 7  |-  ^pm  Fn  ( _V  X.  _V )
3 fndm 5684 . . . . . . 7  |-  (  ^pm  Fn  ( _V  X.  _V )  ->  dom  ^pm  =  ( _V  X.  _V )
)
42, 3ax-mp 5 . . . . . 6  |-  dom  ^pm  =  ( _V  X.  _V )
54ndmov 6458 . . . . 5  |-  ( -.  ( A  e.  _V  /\  C  e.  _V )  ->  ( A  ^pm  C
)  =  (/) )
61, 5nsyl2 130 . . . 4  |-  ( F  e.  ( A  ^pm  C )  ->  ( A  e.  _V  /\  C  e. 
_V ) )
76simpld 460 . . 3  |-  ( F  e.  ( A  ^pm  C )  ->  A  e.  _V )
87adantl 467 . 2  |-  ( ( B  e.  V  /\  F  e.  ( A  ^pm  C ) )  ->  A  e.  _V )
9 simpl 458 . 2  |-  ( ( B  e.  V  /\  F  e.  ( A  ^pm  C ) )  ->  B  e.  V )
10 elpmi 7489 . . . . . 6  |-  ( F  e.  ( A  ^pm  C )  ->  ( F : dom  F --> A  /\  dom  F  C_  C )
)
1110simpld 460 . . . . 5  |-  ( F  e.  ( A  ^pm  C )  ->  F : dom  F --> A )
1211adantl 467 . . . 4  |-  ( ( B  e.  V  /\  F  e.  ( A  ^pm  C ) )  ->  F : dom  F --> A )
13 inss1 3679 . . . 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 666 . . 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 5128 . . . . . 6  |-  ( ( F  |`  dom  F )  |`  B )  =  ( F  |`  ( dom  F  i^i  B ) )
18 funrel 5609 . . . . . . 7  |-  ( Fun 
F  ->  Rel  F )
19 resdm 5157 . . . . . . 7  |-  ( Rel 
F  ->  ( F  |` 
dom  F )  =  F )
20 reseq1 5110 . . . . . . 7  |-  ( ( F  |`  dom  F )  =  F  ->  (
( F  |`  dom  F
)  |`  B )  =  ( F  |`  B ) )
2118, 19, 203syl 18 . . . . . 6  |-  ( Fun 
F  ->  ( ( F  |`  dom  F )  |`  B )  =  ( F  |`  B )
)
2217, 21syl5eqr 2475 . . . . 5  |-  ( Fun 
F  ->  ( F  |`  ( dom  F  i^i  B ) )  =  ( F  |`  B )
)
2312, 16, 223syl 18 . . . 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 213 . 2  |-  ( ( B  e.  V  /\  F  e.  ( A  ^pm  C ) )  -> 
( F  |`  B ) : ( dom  F  i^i  B ) --> A )
26 inss2 3680 . . 3  |-  ( dom 
F  i^i  B )  C_  B
27 elpm2r 7488 . . 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 688 . 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 1263 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 370    = wceq 1437    e. wcel 1867   _Vcvv 3078    i^i cin 3432    C_ wss 3433   (/)c0 3758    X. cxp 4843   dom cdm 4845    |` cres 4847   Rel wrel 4850   Fun wfun 5586    Fn wfn 5587   -->wf 5588  (class class class)co 6296    ^pm cpm 7472
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-8 1869  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-sep 4539  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6588
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-ral 2778  df-rex 2779  df-rab 2782  df-v 3080  df-sbc 3297  df-csb 3393  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-nul 3759  df-if 3907  df-pw 3978  df-sn 3994  df-pr 3996  df-op 4000  df-uni 4214  df-iun 4295  df-br 4418  df-opab 4476  df-mpt 4477  df-id 4760  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5556  df-fun 5594  df-fn 5595  df-f 5596  df-fv 5600  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-1st 6798  df-2nd 6799  df-pm 7474
This theorem is referenced by:  lmres  20253  mbfres  22507  dvnres  22792  cpnres  22798  caures  31837
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