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Theorem elpmg 6802
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 6799 . . . . 5  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( A  ^pm  B
)  =  { g  e.  ~P ( B  X.  A )  |  Fun  g } )
21eleq2d 2363 . . . 4  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( C  e.  ( A  ^pm  B )  <->  C  e.  { g  e. 
~P ( B  X.  A )  |  Fun  g } ) )
3 funeq 5290 . . . . 5  |-  ( g  =  C  ->  ( Fun  g  <->  Fun  C ) )
43elrab 2936 . . . 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 2809 . . . . 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 4816 . . . . . 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 4176 . . . . . 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 3645 . . . . 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 1696   {crab 2560   _Vcvv 2801    C_ wss 3165   ~Pcpw 3638    X. cxp 4703   Fun wfun 5265  (class class class)co 5874    ^pm cpm 6789
This theorem is referenced by:  elpm2g  6803  pmss12g  6810  elpm  6814  pmsspw  6818  lmfss  17040  lmmbr2  18701  iscau2  18719  caussi  18739  causs  18740
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-iota 5235  df-fun 5273  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-pm 6791
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