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Theorem pmresg 6970
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 3569 . . . . 5  |-  ( F  e.  ( A  ^pm  C )  ->  -.  ( A  ^pm  C )  =  (/) )
2 fnpm 6955 . . . . . . 7  |-  ^pm  Fn  ( _V  X.  _V )
3 fndm 5477 . . . . . . 7  |-  (  ^pm  Fn  ( _V  X.  _V )  ->  dom  ^pm  =  ( _V  X.  _V )
)
42, 3ax-mp 8 . . . . . 6  |-  dom  ^pm  =  ( _V  X.  _V )
54ndmov 6163 . . . . 5  |-  ( -.  ( A  e.  _V  /\  C  e.  _V )  ->  ( A  ^pm  C
)  =  (/) )
61, 5nsyl2 121 . . . 4  |-  ( F  e.  ( A  ^pm  C )  ->  ( A  e.  _V  /\  C  e. 
_V ) )
76simpld 446 . . 3  |-  ( F  e.  ( A  ^pm  C )  ->  A  e.  _V )
87adantl 453 . 2  |-  ( ( B  e.  V  /\  F  e.  ( A  ^pm  C ) )  ->  A  e.  _V )
9 simpl 444 . 2  |-  ( ( B  e.  V  /\  F  e.  ( A  ^pm  C ) )  ->  B  e.  V )
10 elpmi 6964 . . . . . 6  |-  ( F  e.  ( A  ^pm  C )  ->  ( F : dom  F --> A  /\  dom  F  C_  C )
)
1110simpld 446 . . . . 5  |-  ( F  e.  ( A  ^pm  C )  ->  F : dom  F --> A )
1211adantl 453 . . . 4  |-  ( ( B  e.  V  /\  F  e.  ( A  ^pm  C ) )  ->  F : dom  F --> A )
13 inss1 3497 . . . 4  |-  ( dom 
F  i^i  B )  C_ 
dom  F
14 fssres 5543 . . . 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 644 . . 3  |-  ( ( B  e.  V  /\  F  e.  ( A  ^pm  C ) )  -> 
( F  |`  ( dom  F  i^i  B ) ) : ( dom 
F  i^i  B ) --> A )
16 ffun 5526 . . . . 5  |-  ( F : dom  F --> A  ->  Fun  F )
17 resres 5092 . . . . . 6  |-  ( ( F  |`  dom  F )  |`  B )  =  ( F  |`  ( dom  F  i^i  B ) )
18 funrel 5404 . . . . . . 7  |-  ( Fun 
F  ->  Rel  F )
19 resdm 5117 . . . . . . 7  |-  ( Rel 
F  ->  ( F  |` 
dom  F )  =  F )
20 reseq1 5073 . . . . . . 7  |-  ( ( F  |`  dom  F )  =  F  ->  (
( F  |`  dom  F
)  |`  B )  =  ( F  |`  B ) )
2118, 19, 203syl 19 . . . . . 6  |-  ( Fun 
F  ->  ( ( F  |`  dom  F )  |`  B )  =  ( F  |`  B )
)
2217, 21syl5eqr 2426 . . . . 5  |-  ( Fun 
F  ->  ( F  |`  ( dom  F  i^i  B ) )  =  ( F  |`  B )
)
2312, 16, 223syl 19 . . . 4  |-  ( ( B  e.  V  /\  F  e.  ( A  ^pm  C ) )  -> 
( F  |`  ( dom  F  i^i  B ) )  =  ( F  |`  B ) )
2423feq1d 5513 . . 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 202 . 2  |-  ( ( B  e.  V  /\  F  e.  ( A  ^pm  C ) )  -> 
( F  |`  B ) : ( dom  F  i^i  B ) --> A )
26 inss2 3498 . . 3  |-  ( dom 
F  i^i  B )  C_  B
27 elpm2r 6963 . . 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 666 . 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 1183 1  |-  ( ( B  e.  V  /\  F  e.  ( A  ^pm  C ) )  -> 
( F  |`  B )  e.  ( A  ^pm  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1717   _Vcvv 2892    i^i cin 3255    C_ wss 3256   (/)c0 3564    X. cxp 4809   dom cdm 4811    |` cres 4813   Rel wrel 4816   Fun wfun 5381    Fn wfn 5382   -->wf 5383  (class class class)co 6013    ^pm cpm 6948
This theorem is referenced by:  lmres  17279  mbfres  19396  dvnres  19677  cpnres  19683  caures  26150
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337  ax-un 4634
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-ral 2647  df-rex 2648  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-op 3759  df-uni 3951  df-iun 4030  df-br 4147  df-opab 4201  df-mpt 4202  df-id 4432  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-fv 5395  df-ov 6016  df-oprab 6017  df-mpt2 6018  df-1st 6281  df-2nd 6282  df-pm 6950
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