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Theorem elpmi 6994
Description: A partial function is a function. (Contributed by Mario Carneiro, 15-Sep-2015.)
Assertion
Ref Expression
elpmi  |-  ( F  e.  ( A  ^pm  B )  ->  ( F : dom  F --> A  /\  dom  F  C_  B )
)

Proof of Theorem elpmi
StepHypRef Expression
1 n0i 3593 . . . 4  |-  ( F  e.  ( A  ^pm  B )  ->  -.  ( A  ^pm  B )  =  (/) )
2 fnpm 6985 . . . . . 6  |-  ^pm  Fn  ( _V  X.  _V )
3 fndm 5503 . . . . . 6  |-  (  ^pm  Fn  ( _V  X.  _V )  ->  dom  ^pm  =  ( _V  X.  _V )
)
42, 3ax-mp 8 . . . . 5  |-  dom  ^pm  =  ( _V  X.  _V )
54ndmov 6190 . . . 4  |-  ( -.  ( A  e.  _V  /\  B  e.  _V )  ->  ( A  ^pm  B
)  =  (/) )
61, 5nsyl2 121 . . 3  |-  ( F  e.  ( A  ^pm  B )  ->  ( A  e.  _V  /\  B  e. 
_V ) )
7 elpm2g 6992 . . 3  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  ( F  e.  ( A  ^pm  B )  <->  ( F : dom  F --> A  /\  dom  F  C_  B ) ) )
86, 7syl 16 . 2  |-  ( F  e.  ( A  ^pm  B )  ->  ( F  e.  ( A  ^pm  B
)  <->  ( F : dom  F --> A  /\  dom  F 
C_  B ) ) )
98ibi 233 1  |-  ( F  e.  ( A  ^pm  B )  ->  ( F : dom  F --> A  /\  dom  F  C_  B )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1721   _Vcvv 2916    C_ wss 3280   (/)c0 3588    X. cxp 4835   dom cdm 4837    Fn wfn 5408   -->wf 5409  (class class class)co 6040    ^pm cpm 6978
This theorem is referenced by:  pmfun  6995  pmresg  7000  equivcau  19206  dvn2bss  19769
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-pm 6980
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