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Theorem pmresg 7033
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 3625 . . . . 5  |-  ( F  e.  ( A  ^pm  C )  ->  -.  ( A  ^pm  C )  =  (/) )
2 fnpm 7018 . . . . . . 7  |-  ^pm  Fn  ( _V  X.  _V )
3 fndm 5536 . . . . . . 7  |-  (  ^pm  Fn  ( _V  X.  _V )  ->  dom  ^pm  =  ( _V  X.  _V )
)
42, 3ax-mp 8 . . . . . 6  |-  dom  ^pm  =  ( _V  X.  _V )
54ndmov 6223 . . . . 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 7027 . . . . . 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 3553 . . . 4  |-  ( dom 
F  i^i  B )  C_ 
dom  F
14 fssres 5602 . . . 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 5585 . . . . 5  |-  ( F : dom  F --> A  ->  Fun  F )
17 resres 5151 . . . . . 6  |-  ( ( F  |`  dom  F )  |`  B )  =  ( F  |`  ( dom  F  i^i  B ) )
18 funrel 5463 . . . . . . 7  |-  ( Fun 
F  ->  Rel  F )
19 resdm 5176 . . . . . . 7  |-  ( Rel 
F  ->  ( F  |` 
dom  F )  =  F )
20 reseq1 5132 . . . . . . 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 2481 . . . . 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 5572 . . 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 3554 . . 3  |-  ( dom 
F  i^i  B )  C_  B
27 elpm2r 7026 . . 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 1652    e. wcel 1725   _Vcvv 2948    i^i cin 3311    C_ wss 3312   (/)c0 3620    X. cxp 4868   dom cdm 4870    |` cres 4872   Rel wrel 4875   Fun wfun 5440    Fn wfn 5441   -->wf 5442  (class class class)co 6073    ^pm cpm 7011
This theorem is referenced by:  lmres  17354  mbfres  19526  dvnres  19807  cpnres  19813  caures  26420
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-pm 7013
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