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Theorem pmresg 6387
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 6362 . . . 4  |-  ^pm  =  ( x  e.  _V ,  y  e.  _V  |->  { f  e.  ~P ( y  X.  x
)  |  Fun  f } )
21elmpt2cl1 5802 . . 3  |-  ( F  e.  ( A  ^pm  C )  ->  A  e.  _V )
32adantl 271 . 2  |-  ( ( B  e.  V  /\  F  e.  ( A  ^pm  C ) )  ->  A  e.  _V )
4 simpl 107 . 2  |-  ( ( B  e.  V  /\  F  e.  ( A  ^pm  C ) )  ->  B  e.  V )
5 elpmi 6378 . . . . . 6  |-  ( F  e.  ( A  ^pm  C )  ->  ( F : dom  F --> A  /\  dom  F  C_  C )
)
65simpld 110 . . . . 5  |-  ( F  e.  ( A  ^pm  C )  ->  F : dom  F --> A )
76adantl 271 . . . 4  |-  ( ( B  e.  V  /\  F  e.  ( A  ^pm  C ) )  ->  F : dom  F --> A )
8 inss1 3209 . . . 4  |-  ( dom 
F  i^i  B )  C_ 
dom  F
9 fssres 5152 . . . 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 404 . . 3  |-  ( ( B  e.  V  /\  F  e.  ( A  ^pm  C ) )  -> 
( F  |`  ( dom  F  i^i  B ) ) : ( dom 
F  i^i  B ) --> A )
11 ffun 5131 . . . . 5  |-  ( F : dom  F --> A  ->  Fun  F )
12 resres 4695 . . . . . 6  |-  ( ( F  |`  dom  F )  |`  B )  =  ( F  |`  ( dom  F  i^i  B ) )
13 funrel 5000 . . . . . . 7  |-  ( Fun 
F  ->  Rel  F )
14 resdm 4720 . . . . . . 7  |-  ( Rel 
F  ->  ( F  |` 
dom  F )  =  F )
15 reseq1 4677 . . . . . . 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, 16syl5eqr 2131 . . . . 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 5116 . . 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 145 . 2  |-  ( ( B  e.  V  /\  F  e.  ( A  ^pm  C ) )  -> 
( F  |`  B ) : ( dom  F  i^i  B ) --> A )
21 inss2 3210 . . 3  |-  ( dom 
F  i^i  B )  C_  B
22 elpm2r 6377 . . 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 429 . 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 1171 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 102    = wceq 1287    e. wcel 1436   {crab 2359   _Vcvv 2615    i^i cin 2987    C_ wss 2988   ~Pcpw 3415    X. cxp 4411   dom cdm 4413    |` cres 4415   Rel wrel 4418   Fun wfun 4977   -->wf 4979  (class class class)co 5615    ^pm cpm 6360
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-13 1447  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-sep 3934  ax-pow 3986  ax-pr 4012  ax-un 4236  ax-setind 4328
This theorem depends on definitions:  df-bi 115  df-3an 924  df-tru 1290  df-fal 1293  df-nf 1393  df-sb 1690  df-eu 1948  df-mo 1949  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ne 2252  df-ral 2360  df-rex 2361  df-rab 2364  df-v 2617  df-sbc 2830  df-dif 2990  df-un 2992  df-in 2994  df-ss 3001  df-pw 3417  df-sn 3437  df-pr 3438  df-op 3440  df-uni 3639  df-br 3823  df-opab 3877  df-id 4096  df-xp 4419  df-rel 4420  df-cnv 4421  df-co 4422  df-dm 4423  df-rn 4424  df-res 4425  df-iota 4948  df-fun 4985  df-fn 4986  df-f 4987  df-fv 4991  df-ov 5618  df-oprab 5619  df-mpt2 5620  df-pm 6362
This theorem is referenced by: (None)
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