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Theorem elpm2r 7035
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 5596 . . . . . . 7  |-  ( F : C --> A  ->  dom  F  =  C )
21feq2d 5582 . . . . . 6  |-  ( F : C --> A  -> 
( F : dom  F --> A  <->  F : C --> A ) )
31sseq1d 3376 . . . . . 6  |-  ( F : C --> A  -> 
( dom  F  C_  B  <->  C 
C_  B ) )
42, 3anbi12d 693 . . . . 5  |-  ( F : C --> A  -> 
( ( F : dom  F --> A  /\  dom  F 
C_  B )  <->  ( F : C --> A  /\  C  C_  B ) ) )
54adantr 453 . . . 4  |-  ( ( F : C --> A  /\  C  C_  B )  -> 
( ( F : dom  F --> A  /\  dom  F 
C_  B )  <->  ( F : C --> A  /\  C  C_  B ) ) )
65ibir 235 . . 3  |-  ( ( F : C --> A  /\  C  C_  B )  -> 
( F : dom  F --> A  /\  dom  F  C_  B ) )
7 elpm2g 7034 . . 3  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( F  e.  ( A  ^pm  B )  <->  ( F : dom  F --> A  /\  dom  F  C_  B ) ) )
86, 7syl5ibr 214 . 2  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( ( F : C
--> A  /\  C  C_  B )  ->  F  e.  ( A  ^pm  B
) ) )
98imp 420 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 178    /\ wa 360    e. wcel 1726    C_ wss 3321   dom cdm 4879   -->wf 5451  (class class class)co 6082    ^pm cpm 7020
This theorem is referenced by:  fpmg  7040  pmresg  7042  rlim  12290  ello12  12311  elo12  12322  sscpwex  14016  catcfuccl  14265  catcxpccl  14305  lmbrf  17325  cnextfval  18094  lmmbrf  19216  iscauf  19234  caucfil  19237  cmetcaulem  19242  lmclimf  19257  ismbf  19523  ismbfcn  19524  mbfconst  19528  cncombf  19551  cnmbf  19552  limcfval  19760  dvfval  19785  dvnff  19810  dvn2bss  19817  dvnfre  19839  taylfvallem1  20274  taylfval  20276  tayl0  20279  taylplem1  20280  taylply2  20285  taylply  20286  dvtaylp  20287  dvntaylp  20288  dvntaylp0  20289  taylthlem1  20290  taylthlem2  20291  ulmval  20297  ulmpm  20300  iseupa  21688  esumcvg  24477
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-sep 4331  ax-nul 4339  ax-pow 4378  ax-pr 4404  ax-un 4702
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-ral 2711  df-rex 2712  df-rab 2715  df-v 2959  df-sbc 3163  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3630  df-if 3741  df-pw 3802  df-sn 3821  df-pr 3822  df-op 3824  df-uni 4017  df-br 4214  df-opab 4268  df-id 4499  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-rn 4890  df-iota 5419  df-fun 5457  df-fn 5458  df-f 5459  df-fv 5463  df-ov 6085  df-oprab 6086  df-mpt2 6087  df-pm 7022
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