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Theorem pmresg 6811
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 3473 . . . . 5  |-  ( F  e.  ( A  ^pm  C )  ->  -.  ( A  ^pm  C )  =  (/) )
2 fnpm 6796 . . . . . . 7  |-  ^pm  Fn  ( _V  X.  _V )
3 fndm 5359 . . . . . . 7  |-  (  ^pm  Fn  ( _V  X.  _V )  ->  dom  ^pm  =  ( _V  X.  _V )
)
42, 3ax-mp 8 . . . . . 6  |-  dom  ^pm  =  ( _V  X.  _V )
54ndmov 6020 . . . . 5  |-  ( -.  ( A  e.  _V  /\  C  e.  _V )  ->  ( A  ^pm  C
)  =  (/) )
61, 5nsyl2 119 . . . 4  |-  ( F  e.  ( A  ^pm  C )  ->  ( A  e.  _V  /\  C  e. 
_V ) )
76simpld 445 . . 3  |-  ( F  e.  ( A  ^pm  C )  ->  A  e.  _V )
87adantl 452 . 2  |-  ( ( B  e.  V  /\  F  e.  ( A  ^pm  C ) )  ->  A  e.  _V )
9 simpl 443 . 2  |-  ( ( B  e.  V  /\  F  e.  ( A  ^pm  C ) )  ->  B  e.  V )
10 elpmi 6805 . . . . . 6  |-  ( F  e.  ( A  ^pm  C )  ->  ( F : dom  F --> A  /\  dom  F  C_  C )
)
1110simpld 445 . . . . 5  |-  ( F  e.  ( A  ^pm  C )  ->  F : dom  F --> A )
1211adantl 452 . . . 4  |-  ( ( B  e.  V  /\  F  e.  ( A  ^pm  C ) )  ->  F : dom  F --> A )
13 inss1 3402 . . . 4  |-  ( dom 
F  i^i  B )  C_ 
dom  F
14 fssres 5424 . . . 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 643 . . 3  |-  ( ( B  e.  V  /\  F  e.  ( A  ^pm  C ) )  -> 
( F  |`  ( dom  F  i^i  B ) ) : ( dom 
F  i^i  B ) --> A )
16 ffun 5407 . . . . 5  |-  ( F : dom  F --> A  ->  Fun  F )
17 resres 4984 . . . . . 6  |-  ( ( F  |`  dom  F )  |`  B )  =  ( F  |`  ( dom  F  i^i  B ) )
18 funrel 5288 . . . . . . 7  |-  ( Fun 
F  ->  Rel  F )
19 resdm 5009 . . . . . . 7  |-  ( Rel 
F  ->  ( F  |` 
dom  F )  =  F )
20 reseq1 4965 . . . . . . 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 2342 . . . . 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 5395 . . 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 201 . 2  |-  ( ( B  e.  V  /\  F  e.  ( A  ^pm  C ) )  -> 
( F  |`  B ) : ( dom  F  i^i  B ) --> A )
26 inss2 3403 . . 3  |-  ( dom 
F  i^i  B )  C_  B
27 elpm2r 6804 . . 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 665 . 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 1181 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 358    = wceq 1632    e. wcel 1696   _Vcvv 2801    i^i cin 3164    C_ wss 3165   (/)c0 3468    X. cxp 4703   dom cdm 4705    |` cres 4707   Rel wrel 4710   Fun wfun 5265    Fn wfn 5266   -->wf 5267  (class class class)co 5874    ^pm cpm 6789
This theorem is referenced by:  lmres  17044  mbfres  19015  dvnres  19296  cpnres  19302  caures  26579
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-pm 6791
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