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Theorem elpm2 7462
Description: The predicate "is a partial function." (Contributed by NM, 15-Nov-2007.) (Revised by Mario Carneiro, 31-Dec-2013.)
Hypotheses
Ref Expression
elmap.1  |-  A  e. 
_V
elmap.2  |-  B  e. 
_V
Assertion
Ref Expression
elpm2  |-  ( F  e.  ( A  ^pm  B )  <->  ( F : dom  F --> A  /\  dom  F 
C_  B ) )

Proof of Theorem elpm2
StepHypRef Expression
1 elmap.1 . 2  |-  A  e. 
_V
2 elmap.2 . 2  |-  B  e. 
_V
3 elpm2g 7447 . 2  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  ( F  e.  ( A  ^pm  B )  <->  ( F : dom  F --> A  /\  dom  F  C_  B ) ) )
41, 2, 3mp2an 672 1  |-  ( F  e.  ( A  ^pm  B )  <->  ( F : dom  F --> A  /\  dom  F 
C_  B ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369    e. wcel 1767   _Vcvv 3118    C_ wss 3481   dom cdm 5005   -->wf 5590  (class class class)co 6295    ^pm cpm 7433
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-sbc 3337  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-br 4454  df-opab 4512  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-fv 5602  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-pm 7435
This theorem is referenced by:  rlimf  13304  rlimss  13305  lo1f  13321  lo1dm  13322  o1f  13332  o1dm  13333  coapm  15273  pmltpclem2  21729  mbff  21902  limcrcl  22146  dvnres  22202  c1liplem1  22265  c1lip2  22267  ulmf2  22646
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