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Theorem pmresg 6669
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 6644 . . . 4  |-  ^pm  =  ( x  e.  _V ,  y  e.  _V  |->  { f  e.  ~P ( y  X.  x
)  |  Fun  f } )
21elmpocl1 6063 . . 3  |-  ( F  e.  ( A  ^pm  C )  ->  A  e.  _V )
32adantl 277 . 2  |-  ( ( B  e.  V  /\  F  e.  ( A  ^pm  C ) )  ->  A  e.  _V )
4 simpl 109 . 2  |-  ( ( B  e.  V  /\  F  e.  ( A  ^pm  C ) )  ->  B  e.  V )
5 elpmi 6660 . . . . . 6  |-  ( F  e.  ( A  ^pm  C )  ->  ( F : dom  F --> A  /\  dom  F  C_  C )
)
65simpld 112 . . . . 5  |-  ( F  e.  ( A  ^pm  C )  ->  F : dom  F --> A )
76adantl 277 . . . 4  |-  ( ( B  e.  V  /\  F  e.  ( A  ^pm  C ) )  ->  F : dom  F --> A )
8 inss1 3355 . . . 4  |-  ( dom 
F  i^i  B )  C_ 
dom  F
9 fssres 5386 . . . 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 413 . . 3  |-  ( ( B  e.  V  /\  F  e.  ( A  ^pm  C ) )  -> 
( F  |`  ( dom  F  i^i  B ) ) : ( dom 
F  i^i  B ) --> A )
11 ffun 5363 . . . . 5  |-  ( F : dom  F --> A  ->  Fun  F )
12 resres 4914 . . . . . 6  |-  ( ( F  |`  dom  F )  |`  B )  =  ( F  |`  ( dom  F  i^i  B ) )
13 funrel 5228 . . . . . . 7  |-  ( Fun 
F  ->  Rel  F )
14 resdm 4941 . . . . . . 7  |-  ( Rel 
F  ->  ( F  |` 
dom  F )  =  F )
15 reseq1 4896 . . . . . . 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 2224 . . . . 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 5347 . . 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 147 . 2  |-  ( ( B  e.  V  /\  F  e.  ( A  ^pm  C ) )  -> 
( F  |`  B ) : ( dom  F  i^i  B ) --> A )
21 inss2 3356 . . 3  |-  ( dom 
F  i^i  B )  C_  B
22 elpm2r 6659 . . 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 438 . 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 1237 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 104    = wceq 1353    e. wcel 2148   {crab 2459   _Vcvv 2737    i^i cin 3128    C_ wss 3129   ~Pcpw 3574    X. cxp 4620   dom cdm 4622    |` cres 4624   Rel wrel 4627   Fun wfun 5205   -->wf 5207  (class class class)co 5868    ^pm cpm 6642
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171  ax-pr 4205  ax-un 4429  ax-setind 4532
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2739  df-sbc 2963  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-br 4001  df-opab 4062  df-id 4289  df-xp 4628  df-rel 4629  df-cnv 4630  df-co 4631  df-dm 4632  df-rn 4633  df-res 4634  df-iota 5173  df-fun 5213  df-fn 5214  df-f 5215  df-fv 5219  df-ov 5871  df-oprab 5872  df-mpo 5873  df-pm 6644
This theorem is referenced by:  lmres  13381
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