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Theorem pmsspw 6802
Description: Partial maps are a subset of the power set of the cross product of its arguments. (Contributed by Mario Carneiro, 2-Jan-2017.)
Assertion
Ref Expression
pmsspw  |-  ( A 
^pm  B )  C_  ~P ( B  X.  A
)

Proof of Theorem pmsspw
Dummy variable  f is distinct from all other variables.
StepHypRef Expression
1 n0i 3460 . . . . . . 7  |-  ( f  e.  ( A  ^pm  B )  ->  -.  ( A  ^pm  B )  =  (/) )
2 fnpm 6780 . . . . . . . . 9  |-  ^pm  Fn  ( _V  X.  _V )
3 fndm 5343 . . . . . . . . 9  |-  (  ^pm  Fn  ( _V  X.  _V )  ->  dom  ^pm  =  ( _V  X.  _V )
)
42, 3ax-mp 8 . . . . . . . 8  |-  dom  ^pm  =  ( _V  X.  _V )
54ndmov 6004 . . . . . . 7  |-  ( -.  ( A  e.  _V  /\  B  e.  _V )  ->  ( A  ^pm  B
)  =  (/) )
61, 5nsyl2 119 . . . . . 6  |-  ( f  e.  ( A  ^pm  B )  ->  ( A  e.  _V  /\  B  e. 
_V ) )
7 elpmg 6786 . . . . . 6  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  ( f  e.  ( A  ^pm  B )  <->  ( Fun  f  /\  f  C_  ( B  X.  A
) ) ) )
86, 7syl 15 . . . . 5  |-  ( f  e.  ( A  ^pm  B )  ->  ( f  e.  ( A  ^pm  B
)  <->  ( Fun  f  /\  f  C_  ( B  X.  A ) ) ) )
98ibi 232 . . . 4  |-  ( f  e.  ( A  ^pm  B )  ->  ( Fun  f  /\  f  C_  ( B  X.  A ) ) )
109simprd 449 . . 3  |-  ( f  e.  ( A  ^pm  B )  ->  f  C_  ( B  X.  A
) )
11 vex 2791 . . . 4  |-  f  e. 
_V
1211elpw 3631 . . 3  |-  ( f  e.  ~P ( B  X.  A )  <->  f  C_  ( B  X.  A
) )
1310, 12sylibr 203 . 2  |-  ( f  e.  ( A  ^pm  B )  ->  f  e.  ~P ( B  X.  A
) )
1413ssriv 3184 1  |-  ( A 
^pm  B )  C_  ~P ( B  X.  A
)
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   _Vcvv 2788    C_ wss 3152   (/)c0 3455   ~Pcpw 3625    X. cxp 4687   dom cdm 4689   Fun wfun 5249    Fn wfn 5250  (class class class)co 5858    ^pm cpm 6773
This theorem is referenced by:  mapsspw  6803  wunpm  8347
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-pm 6775
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