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Theorem elpm2r 6993
Description: Sufficient condition for being a partial function. (Contributed by NM, 31-Dec-2013.)
Assertion
Ref Expression
elpm2r  |-  ( ( ( A  e.  V  /\  B  e.  W
)  /\  ( F : C --> A  /\  C  C_  B ) )  ->  F  e.  ( A  ^pm  B ) )

Proof of Theorem elpm2r
StepHypRef Expression
1 fdm 5554 . . . . . . 7  |-  ( F : C --> A  ->  dom  F  =  C )
21feq2d 5540 . . . . . 6  |-  ( F : C --> A  -> 
( F : dom  F --> A  <->  F : C --> A ) )
31sseq1d 3335 . . . . . 6  |-  ( F : C --> A  -> 
( dom  F  C_  B  <->  C 
C_  B ) )
42, 3anbi12d 692 . . . . 5  |-  ( F : C --> A  -> 
( ( F : dom  F --> A  /\  dom  F 
C_  B )  <->  ( F : C --> A  /\  C  C_  B ) ) )
54adantr 452 . . . 4  |-  ( ( F : C --> A  /\  C  C_  B )  -> 
( ( F : dom  F --> A  /\  dom  F 
C_  B )  <->  ( F : C --> A  /\  C  C_  B ) ) )
65ibir 234 . . 3  |-  ( ( F : C --> A  /\  C  C_  B )  -> 
( F : dom  F --> A  /\  dom  F  C_  B ) )
7 elpm2g 6992 . . 3  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( F  e.  ( A  ^pm  B )  <->  ( F : dom  F --> A  /\  dom  F  C_  B ) ) )
86, 7syl5ibr 213 . 2  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( ( F : C
--> A  /\  C  C_  B )  ->  F  e.  ( A  ^pm  B
) ) )
98imp 419 1  |-  ( ( ( A  e.  V  /\  B  e.  W
)  /\  ( F : C --> A  /\  C  C_  B ) )  ->  F  e.  ( A  ^pm  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    e. wcel 1721    C_ wss 3280   dom cdm 4837   -->wf 5409  (class class class)co 6040    ^pm cpm 6978
This theorem is referenced by:  fpmg  6998  pmresg  7000  rlim  12244  ello12  12265  elo12  12276  sscpwex  13970  catcfuccl  14219  catcxpccl  14259  lmbrf  17278  cnextfval  18046  lmmbrf  19168  iscauf  19186  caucfil  19189  cmetcaulem  19194  lmclimf  19209  ismbf  19475  ismbfcn  19476  mbfconst  19480  cncombf  19503  cnmbf  19504  limcfval  19712  dvfval  19737  dvnff  19762  dvn2bss  19769  dvnfre  19791  taylfvallem1  20226  taylfval  20228  tayl0  20231  taylplem1  20232  taylply2  20237  taylply  20238  dvtaylp  20239  dvntaylp  20240  dvntaylp0  20241  taylthlem1  20242  taylthlem2  20243  ulmval  20249  ulmpm  20252  iseupa  21640  esumcvg  24429
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-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-br 4173  df-opab 4227  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  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-pm 6980
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