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Theorem elpmg 7034
Description: The predicate "is a partial function." (Contributed by Mario Carneiro, 14-Nov-2013.)
Assertion
Ref Expression
elpmg  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( C  e.  ( A  ^pm  B )  <->  ( Fun  C  /\  C  C_  ( B  X.  A
) ) ) )

Proof of Theorem elpmg
Dummy variable  g is distinct from all other variables.
StepHypRef Expression
1 pmvalg 7031 . . . . 5  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( A  ^pm  B
)  =  { g  e.  ~P ( B  X.  A )  |  Fun  g } )
21eleq2d 2505 . . . 4  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( C  e.  ( A  ^pm  B )  <->  C  e.  { g  e. 
~P ( B  X.  A )  |  Fun  g } ) )
3 funeq 5475 . . . . 5  |-  ( g  =  C  ->  ( Fun  g  <->  Fun  C ) )
43elrab 3094 . . . 4  |-  ( C  e.  { g  e. 
~P ( B  X.  A )  |  Fun  g }  <->  ( C  e. 
~P ( B  X.  A )  /\  Fun  C ) )
52, 4syl6bb 254 . . 3  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( C  e.  ( A  ^pm  B )  <->  ( C  e.  ~P ( B  X.  A )  /\  Fun  C ) ) )
6 ancom 439 . . 3  |-  ( ( C  e.  ~P ( B  X.  A )  /\  Fun  C )  <->  ( Fun  C  /\  C  e.  ~P ( B  X.  A
) ) )
75, 6syl6bb 254 . 2  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( C  e.  ( A  ^pm  B )  <->  ( Fun  C  /\  C  e.  ~P ( B  X.  A ) ) ) )
8 elex 2966 . . . . 5  |-  ( C  e.  ~P ( B  X.  A )  ->  C  e.  _V )
98a1i 11 . . . 4  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( C  e.  ~P ( B  X.  A
)  ->  C  e.  _V ) )
10 xpexg 4991 . . . . . 6  |-  ( ( B  e.  W  /\  A  e.  V )  ->  ( B  X.  A
)  e.  _V )
1110ancoms 441 . . . . 5  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( B  X.  A
)  e.  _V )
12 ssexg 4351 . . . . . 6  |-  ( ( C  C_  ( B  X.  A )  /\  ( B  X.  A )  e. 
_V )  ->  C  e.  _V )
1312expcom 426 . . . . 5  |-  ( ( B  X.  A )  e.  _V  ->  ( C  C_  ( B  X.  A )  ->  C  e.  _V ) )
1411, 13syl 16 . . . 4  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( C  C_  ( B  X.  A )  ->  C  e.  _V )
)
15 elpwg 3808 . . . . 5  |-  ( C  e.  _V  ->  ( C  e.  ~P ( B  X.  A )  <->  C  C_  ( B  X.  A ) ) )
1615a1i 11 . . . 4  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( C  e.  _V  ->  ( C  e.  ~P ( B  X.  A
)  <->  C  C_  ( B  X.  A ) ) ) )
179, 14, 16pm5.21ndd 345 . . 3  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( C  e.  ~P ( B  X.  A
)  <->  C  C_  ( B  X.  A ) ) )
1817anbi2d 686 . 2  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( ( Fun  C  /\  C  e.  ~P ( B  X.  A
) )  <->  ( Fun  C  /\  C  C_  ( B  X.  A ) ) ) )
197, 18bitrd 246 1  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( C  e.  ( A  ^pm  B )  <->  ( Fun  C  /\  C  C_  ( B  X.  A
) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    e. wcel 1726   {crab 2711   _Vcvv 2958    C_ wss 3322   ~Pcpw 3801    X. cxp 4878   Fun wfun 5450  (class class class)co 6083    ^pm cpm 7021
This theorem is referenced by:  elpm2g  7035  pmss12g  7042  elpm  7046  pmsspw  7050  lmfss  17362  lmmbr2  19214  iscau2  19232  caussi  19252  causs  19253
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4215  df-opab 4269  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-iota 5420  df-fun 5458  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-pm 7023
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