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Theorem elpm2 7244
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 7229 . 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 1756   _Vcvv 2972    C_ wss 3328   dom cdm 4840   -->wf 5414  (class class class)co 6091    ^pm cpm 7215
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-rab 2724  df-v 2974  df-sbc 3187  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-op 3884  df-uni 4092  df-br 4293  df-opab 4351  df-id 4636  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-fv 5426  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-pm 7217
This theorem is referenced by:  rlimf  12979  rlimss  12980  lo1f  12996  lo1dm  12997  o1f  13007  o1dm  13008  coapm  14939  pmltpclem2  20933  mbff  21105  limcrcl  21349  dvnres  21405  c1liplem1  21468  c1lip2  21470  ulmf2  21849
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