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Theorem pmsspw 6677
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 6645 . . . . . . 7  |-  ^pm  =  ( x  e.  _V ,  y  e.  _V  |->  { f  e.  ~P ( y  X.  x
)  |  Fun  f } )
21elmpocl 6063 . . . . . 6  |-  ( f  e.  ( A  ^pm  B )  ->  ( A  e.  _V  /\  B  e. 
_V ) )
3 elpmg 6658 . . . . . 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 176 . . . 4  |-  ( f  e.  ( A  ^pm  B )  ->  ( Fun  f  /\  f  C_  ( B  X.  A ) ) )
65simprd 114 . . 3  |-  ( f  e.  ( A  ^pm  B )  ->  f  C_  ( B  X.  A
) )
7 velpw 3581 . . 3  |-  ( f  e.  ~P ( B  X.  A )  <->  f  C_  ( B  X.  A
) )
86, 7sylibr 134 . 2  |-  ( f  e.  ( A  ^pm  B )  ->  f  e.  ~P ( B  X.  A
) )
98ssriv 3159 1  |-  ( A 
^pm  B )  C_  ~P ( B  X.  A
)
Colors of variables: wff set class
Syntax hints:    /\ wa 104    <-> wb 105    e. wcel 2148   {crab 2459   _Vcvv 2737    C_ wss 3129   ~Pcpw 3574    X. cxp 4621   Fun wfun 5206  (class class class)co 5869    ^pm cpm 6643
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 4206  ax-un 4430  ax-setind 4533
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 4290  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-iota 5174  df-fun 5214  df-fv 5220  df-ov 5872  df-oprab 5873  df-mpo 5874  df-pm 6645
This theorem is referenced by:  mapsspw  6678
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