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Theorem pmresg 6642
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
Dummy variables  x  f  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-pm 6617 . . . 4  |-  ^pm  =  ( x  e.  _V ,  y  e.  _V  |->  { f  e.  ~P ( y  X.  x
)  |  Fun  f } )
21elmpocl1 6037 . . 3  |-  ( F  e.  ( A  ^pm  C )  ->  A  e.  _V )
32adantl 275 . 2  |-  ( ( B  e.  V  /\  F  e.  ( A  ^pm  C ) )  ->  A  e.  _V )
4 simpl 108 . 2  |-  ( ( B  e.  V  /\  F  e.  ( A  ^pm  C ) )  ->  B  e.  V )
5 elpmi 6633 . . . . . 6  |-  ( F  e.  ( A  ^pm  C )  ->  ( F : dom  F --> A  /\  dom  F  C_  C )
)
65simpld 111 . . . . 5  |-  ( F  e.  ( A  ^pm  C )  ->  F : dom  F --> A )
76adantl 275 . . . 4  |-  ( ( B  e.  V  /\  F  e.  ( A  ^pm  C ) )  ->  F : dom  F --> A )
8 inss1 3342 . . . 4  |-  ( dom 
F  i^i  B )  C_ 
dom  F
9 fssres 5363 . . . 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 )
107, 8, 9sylancl 410 . . 3  |-  ( ( B  e.  V  /\  F  e.  ( A  ^pm  C ) )  -> 
( F  |`  ( dom  F  i^i  B ) ) : ( dom 
F  i^i  B ) --> A )
11 ffun 5340 . . . . 5  |-  ( F : dom  F --> A  ->  Fun  F )
12 resres 4896 . . . . . 6  |-  ( ( F  |`  dom  F )  |`  B )  =  ( F  |`  ( dom  F  i^i  B ) )
13 funrel 5205 . . . . . . 7  |-  ( Fun 
F  ->  Rel  F )
14 resdm 4923 . . . . . . 7  |-  ( Rel 
F  ->  ( F  |` 
dom  F )  =  F )
15 reseq1 4878 . . . . . . 7  |-  ( ( F  |`  dom  F )  =  F  ->  (
( F  |`  dom  F
)  |`  B )  =  ( F  |`  B ) )
1613, 14, 153syl 17 . . . . . 6  |-  ( Fun 
F  ->  ( ( F  |`  dom  F )  |`  B )  =  ( F  |`  B )
)
1712, 16eqtr3id 2213 . . . . 5  |-  ( Fun 
F  ->  ( F  |`  ( dom  F  i^i  B ) )  =  ( F  |`  B )
)
187, 11, 173syl 17 . . . 4  |-  ( ( B  e.  V  /\  F  e.  ( A  ^pm  C ) )  -> 
( F  |`  ( dom  F  i^i  B ) )  =  ( F  |`  B ) )
1918feq1d 5324 . . 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 ) )
2010, 19mpbid 146 . 2  |-  ( ( B  e.  V  /\  F  e.  ( A  ^pm  C ) )  -> 
( F  |`  B ) : ( dom  F  i^i  B ) --> A )
21 inss2 3343 . . 3  |-  ( dom 
F  i^i  B )  C_  B
22 elpm2r 6632 . . 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 )
)
2321, 22mpanr2 435 . 2  |-  ( ( ( A  e.  _V  /\  B  e.  V )  /\  ( F  |`  B ) : ( dom  F  i^i  B
) --> A )  -> 
( F  |`  B )  e.  ( A  ^pm  B ) )
243, 4, 20, 23syl21anc 1227 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 103    = wceq 1343    e. wcel 2136   {crab 2448   _Vcvv 2726    i^i cin 3115    C_ wss 3116   ~Pcpw 3559    X. cxp 4602   dom cdm 4604    |` cres 4606   Rel wrel 4609   Fun wfun 5182   -->wf 5184  (class class class)co 5842    ^pm cpm 6615
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514
This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-ral 2449  df-rex 2450  df-rab 2453  df-v 2728  df-sbc 2952  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-br 3983  df-opab 4044  df-id 4271  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-fv 5196  df-ov 5845  df-oprab 5846  df-mpo 5847  df-pm 6617
This theorem is referenced by:  lmres  12888
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