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Theorem pmsspw 6713
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 6681 . . . . . . 7  |-  ^pm  =  ( x  e.  _V ,  y  e.  _V  |->  { f  e.  ~P ( y  X.  x
)  |  Fun  f } )
21elmpocl 6095 . . . . . 6  |-  ( f  e.  ( A  ^pm  B )  ->  ( A  e.  _V  /\  B  e. 
_V ) )
3 elpmg 6694 . . . . . 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 3600 . . 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 3174 1  |-  ( A 
^pm  B )  C_  ~P ( B  X.  A
)
Colors of variables: wff set class
Syntax hints:    /\ wa 104    <-> wb 105    e. wcel 2160   {crab 2472   _Vcvv 2752    C_ wss 3144   ~Pcpw 3593    X. cxp 4645   Fun wfun 5232  (class class class)co 5900    ^pm cpm 6679
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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4139  ax-pow 4195  ax-pr 4230  ax-un 4454  ax-setind 4557
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-ral 2473  df-rex 2474  df-rab 2477  df-v 2754  df-sbc 2978  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-pw 3595  df-sn 3616  df-pr 3617  df-op 3619  df-uni 3828  df-br 4022  df-opab 4083  df-id 4314  df-xp 4653  df-rel 4654  df-cnv 4655  df-co 4656  df-dm 4657  df-iota 5199  df-fun 5240  df-fv 5246  df-ov 5903  df-oprab 5904  df-mpo 5905  df-pm 6681
This theorem is referenced by:  mapsspw  6714
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