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Theorem elpm2 6942
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 6930 . 2  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  ( F  e.  ( A  ^pm  B )  <->  ( F : dom  F --> A  /\  dom  F  C_  B ) ) )
41, 2, 3mp2an 653 1  |-  ( F  e.  ( A  ^pm  B )  <->  ( F : dom  F --> A  /\  dom  F 
C_  B ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358    e. wcel 1715   _Vcvv 2873    C_ wss 3238   dom cdm 4792   -->wf 5354  (class class class)co 5981    ^pm cpm 6916
This theorem is referenced by:  rlimf  12182  rlimss  12183  lo1f  12199  lo1dm  12200  o1f  12210  o1dm  12211  coapm  14113  pmltpclem2  19024  mbff  19197  limcrcl  19439  dvnres  19495  c1liplem1  19558  c1lip2  19560  ulmf2  19978
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-sep 4243  ax-nul 4251  ax-pow 4290  ax-pr 4316  ax-un 4615
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-ral 2633  df-rex 2634  df-rab 2637  df-v 2875  df-sbc 3078  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-nul 3544  df-if 3655  df-pw 3716  df-sn 3735  df-pr 3736  df-op 3738  df-uni 3930  df-br 4126  df-opab 4180  df-id 4412  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-rn 4803  df-iota 5322  df-fun 5360  df-fn 5361  df-f 5362  df-fv 5366  df-ov 5984  df-oprab 5985  df-mpt2 5986  df-pm 6918
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