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Theorem elpmg 6786
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 6783 . . . . 5  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( A  ^pm  B
)  =  { g  e.  ~P ( B  X.  A )  |  Fun  g } )
21eleq2d 2350 . . . 4  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( C  e.  ( A  ^pm  B )  <->  C  e.  { g  e. 
~P ( B  X.  A )  |  Fun  g } ) )
3 funeq 5274 . . . . 5  |-  ( g  =  C  ->  ( Fun  g  <->  Fun  C ) )
43elrab 2923 . . . 4  |-  ( C  e.  { g  e. 
~P ( B  X.  A )  |  Fun  g }  <->  ( C  e. 
~P ( B  X.  A )  /\  Fun  C ) )
52, 4syl6bb 252 . . 3  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( C  e.  ( A  ^pm  B )  <->  ( C  e.  ~P ( B  X.  A )  /\  Fun  C ) ) )
6 ancom 437 . . 3  |-  ( ( C  e.  ~P ( B  X.  A )  /\  Fun  C )  <->  ( Fun  C  /\  C  e.  ~P ( B  X.  A
) ) )
75, 6syl6bb 252 . 2  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( C  e.  ( A  ^pm  B )  <->  ( Fun  C  /\  C  e.  ~P ( B  X.  A ) ) ) )
8 elex 2796 . . . . 5  |-  ( C  e.  ~P ( B  X.  A )  ->  C  e.  _V )
98a1i 10 . . . 4  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( C  e.  ~P ( B  X.  A
)  ->  C  e.  _V ) )
10 xpexg 4800 . . . . . 6  |-  ( ( B  e.  W  /\  A  e.  V )  ->  ( B  X.  A
)  e.  _V )
1110ancoms 439 . . . . 5  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( B  X.  A
)  e.  _V )
12 ssexg 4160 . . . . . 6  |-  ( ( C  C_  ( B  X.  A )  /\  ( B  X.  A )  e. 
_V )  ->  C  e.  _V )
1312expcom 424 . . . . 5  |-  ( ( B  X.  A )  e.  _V  ->  ( C  C_  ( B  X.  A )  ->  C  e.  _V ) )
1411, 13syl 15 . . . 4  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( C  C_  ( B  X.  A )  ->  C  e.  _V )
)
15 elpwg 3632 . . . . 5  |-  ( C  e.  _V  ->  ( C  e.  ~P ( B  X.  A )  <->  C  C_  ( B  X.  A ) ) )
1615a1i 10 . . . 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 343 . . 3  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( C  e.  ~P ( B  X.  A
)  <->  C  C_  ( B  X.  A ) ) )
1817anbi2d 684 . 2  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( ( Fun  C  /\  C  e.  ~P ( B  X.  A
) )  <->  ( Fun  C  /\  C  C_  ( B  X.  A ) ) ) )
197, 18bitrd 244 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 176    /\ wa 358    e. wcel 1684   {crab 2547   _Vcvv 2788    C_ wss 3152   ~Pcpw 3625    X. cxp 4687   Fun wfun 5249  (class class class)co 5858    ^pm cpm 6773
This theorem is referenced by:  elpm2g  6787  pmss12g  6794  elpm  6798  pmsspw  6802  lmfss  17024  lmmbr2  18685  iscau2  18703  caussi  18723  causs  18724
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-iota 5219  df-fun 5257  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-pm 6775
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