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Theorem pmsspw 6480
Description: Partial maps are a subset of the power set of the Cartesian 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 variables  f  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-pm 6448 . . . . . . 7  |-  ^pm  =  ( x  e.  _V ,  y  e.  _V  |->  { f  e.  ~P ( y  X.  x
)  |  Fun  f } )
21elmpt2cl 5880 . . . . . 6  |-  ( f  e.  ( A  ^pm  B )  ->  ( A  e.  _V  /\  B  e. 
_V ) )
3 elpmg 6461 . . . . . 6  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  ( f  e.  ( A  ^pm  B )  <->  ( Fun  f  /\  f  C_  ( B  X.  A
) ) ) )
42, 3syl 14 . . . . 5  |-  ( f  e.  ( A  ^pm  B )  ->  ( f  e.  ( A  ^pm  B
)  <->  ( Fun  f  /\  f  C_  ( B  X.  A ) ) ) )
54ibi 175 . . . 4  |-  ( f  e.  ( A  ^pm  B )  ->  ( Fun  f  /\  f  C_  ( B  X.  A ) ) )
65simprd 113 . . 3  |-  ( f  e.  ( A  ^pm  B )  ->  f  C_  ( B  X.  A
) )
7 selpw 3456 . . 3  |-  ( f  e.  ~P ( B  X.  A )  <->  f  C_  ( B  X.  A
) )
86, 7sylibr 133 . 2  |-  ( f  e.  ( A  ^pm  B )  ->  f  e.  ~P ( B  X.  A
) )
98ssriv 3043 1  |-  ( A 
^pm  B )  C_  ~P ( B  X.  A
)
Colors of variables: wff set class
Syntax hints:    /\ wa 103    <-> wb 104    e. wcel 1445   {crab 2374   _Vcvv 2633    C_ wss 3013   ~Pcpw 3449    X. cxp 4465   Fun wfun 5043  (class class class)co 5690    ^pm cpm 6446
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 582  ax-in2 583  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-13 1456  ax-14 1457  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077  ax-sep 3978  ax-pow 4030  ax-pr 4060  ax-un 4284  ax-setind 4381
This theorem depends on definitions:  df-bi 116  df-3an 929  df-tru 1299  df-fal 1302  df-nf 1402  df-sb 1700  df-eu 1958  df-mo 1959  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ne 2263  df-ral 2375  df-rex 2376  df-rab 2379  df-v 2635  df-sbc 2855  df-dif 3015  df-un 3017  df-in 3019  df-ss 3026  df-pw 3451  df-sn 3472  df-pr 3473  df-op 3475  df-uni 3676  df-br 3868  df-opab 3922  df-id 4144  df-xp 4473  df-rel 4474  df-cnv 4475  df-co 4476  df-dm 4477  df-iota 5014  df-fun 5051  df-fv 5057  df-ov 5693  df-oprab 5694  df-mpt2 5695  df-pm 6448
This theorem is referenced by:  mapsspw  6481
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