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Theorem pmsspw 7446
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 variable  f is distinct from all other variables.
StepHypRef Expression
1 n0i 3788 . . . . . . 7  |-  ( f  e.  ( A  ^pm  B )  ->  -.  ( A  ^pm  B )  =  (/) )
2 fnpm 7420 . . . . . . . . 9  |-  ^pm  Fn  ( _V  X.  _V )
3 fndm 5662 . . . . . . . . 9  |-  (  ^pm  Fn  ( _V  X.  _V )  ->  dom  ^pm  =  ( _V  X.  _V )
)
42, 3ax-mp 5 . . . . . . . 8  |-  dom  ^pm  =  ( _V  X.  _V )
54ndmov 6432 . . . . . . 7  |-  ( -.  ( A  e.  _V  /\  B  e.  _V )  ->  ( A  ^pm  B
)  =  (/) )
61, 5nsyl2 127 . . . . . 6  |-  ( f  e.  ( A  ^pm  B )  ->  ( A  e.  _V  /\  B  e. 
_V ) )
7 elpmg 7427 . . . . . 6  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  ( f  e.  ( A  ^pm  B )  <->  ( Fun  f  /\  f  C_  ( B  X.  A
) ) ) )
86, 7syl 16 . . . . 5  |-  ( f  e.  ( A  ^pm  B )  ->  ( f  e.  ( A  ^pm  B
)  <->  ( Fun  f  /\  f  C_  ( B  X.  A ) ) ) )
98ibi 241 . . . 4  |-  ( f  e.  ( A  ^pm  B )  ->  ( Fun  f  /\  f  C_  ( B  X.  A ) ) )
109simprd 461 . . 3  |-  ( f  e.  ( A  ^pm  B )  ->  f  C_  ( B  X.  A
) )
11 selpw 4006 . . 3  |-  ( f  e.  ~P ( B  X.  A )  <->  f  C_  ( B  X.  A
) )
1210, 11sylibr 212 . 2  |-  ( f  e.  ( A  ^pm  B )  ->  f  e.  ~P ( B  X.  A
) )
1312ssriv 3493 1  |-  ( A 
^pm  B )  C_  ~P ( B  X.  A
)
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 367    = wceq 1398    e. wcel 1823   _Vcvv 3106    C_ wss 3461   (/)c0 3783   ~Pcpw 3999    X. cxp 4986   dom cdm 4988   Fun wfun 5564    Fn wfn 5565  (class class class)co 6270    ^pm cpm 7413
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-fv 5578  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-1st 6773  df-2nd 6774  df-pm 7415
This theorem is referenced by:  mapsspw  7447  wunpm  9092
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