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Theorem elpmg 6657
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 6652 . . . . 5  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( A  ^pm  B
)  =  { g  e.  ~P ( B  X.  A )  |  Fun  g } )
21eleq2d 2247 . . . 4  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( C  e.  ( A  ^pm  B )  <->  C  e.  { g  e. 
~P ( B  X.  A )  |  Fun  g } ) )
3 funeq 5231 . . . . 5  |-  ( g  =  C  ->  ( Fun  g  <->  Fun  C ) )
43elrab 2893 . . . 4  |-  ( C  e.  { g  e. 
~P ( B  X.  A )  |  Fun  g }  <->  ( C  e. 
~P ( B  X.  A )  /\  Fun  C ) )
52, 4bitrdi 196 . . 3  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( C  e.  ( A  ^pm  B )  <->  ( C  e.  ~P ( B  X.  A )  /\  Fun  C ) ) )
6 ancom 266 . . 3  |-  ( ( C  e.  ~P ( B  X.  A )  /\  Fun  C )  <->  ( Fun  C  /\  C  e.  ~P ( B  X.  A
) ) )
75, 6bitrdi 196 . 2  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( C  e.  ( A  ^pm  B )  <->  ( Fun  C  /\  C  e.  ~P ( B  X.  A ) ) ) )
8 elex 2748 . . . . 5  |-  ( C  e.  ~P ( B  X.  A )  ->  C  e.  _V )
98a1i 9 . . . 4  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( C  e.  ~P ( B  X.  A
)  ->  C  e.  _V ) )
10 xpexg 4736 . . . . . 6  |-  ( ( B  e.  W  /\  A  e.  V )  ->  ( B  X.  A
)  e.  _V )
1110ancoms 268 . . . . 5  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( B  X.  A
)  e.  _V )
12 ssexg 4139 . . . . . 6  |-  ( ( C  C_  ( B  X.  A )  /\  ( B  X.  A )  e. 
_V )  ->  C  e.  _V )
1312expcom 116 . . . . 5  |-  ( ( B  X.  A )  e.  _V  ->  ( C  C_  ( B  X.  A )  ->  C  e.  _V ) )
1411, 13syl 14 . . . 4  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( C  C_  ( B  X.  A )  ->  C  e.  _V )
)
15 elpwg 3582 . . . . 5  |-  ( C  e.  _V  ->  ( C  e.  ~P ( B  X.  A )  <->  C  C_  ( B  X.  A ) ) )
1615a1i 9 . . . 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 705 . . 3  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( C  e.  ~P ( B  X.  A
)  <->  C  C_  ( B  X.  A ) ) )
1817anbi2d 464 . 2  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( ( Fun  C  /\  C  e.  ~P ( B  X.  A
) )  <->  ( Fun  C  /\  C  C_  ( B  X.  A ) ) ) )
197, 18bitrd 188 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    /\ wa 104    <-> wb 105    e. wcel 2148   {crab 2459   _Vcvv 2737    C_ wss 3129   ~Pcpw 3574    X. cxp 4620   Fun wfun 5205  (class class class)co 5868    ^pm cpm 6642
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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171  ax-pr 4205  ax-un 4429  ax-setind 4532
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2739  df-sbc 2963  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-br 4001  df-opab 4062  df-id 4289  df-xp 4628  df-rel 4629  df-cnv 4630  df-co 4631  df-dm 4632  df-iota 5173  df-fun 5213  df-fv 5219  df-ov 5871  df-oprab 5872  df-mpo 5873  df-pm 6644
This theorem is referenced by:  elpm2g  6658  pmss12g  6668  elpm  6672  pmsspw  6676  ennnfonelemj0  12372  lmfss  13377
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