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Theorem bafval 21160
Description: Value of the function for the base set of a normed complex vector space. (Contributed by NM, 23-Apr-2007.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
bafval.1  |-  X  =  ( BaseSet `  U )
bafval.2  |-  G  =  ( +v `  U
)
Assertion
Ref Expression
bafval  |-  X  =  ran  G

Proof of Theorem bafval
Dummy variable  u is distinct from all other variables.
StepHypRef Expression
1 fveq2 5525 . . . . 5  |-  ( u  =  U  ->  ( +v `  u )  =  ( +v `  U
) )
21rneqd 4906 . . . 4  |-  ( u  =  U  ->  ran  ( +v `  u )  =  ran  ( +v
`  U ) )
3 df-ba 21152 . . . 4  |-  BaseSet  =  ( u  e.  _V  |->  ran  ( +v `  u
) )
4 fvex 5539 . . . . 5  |-  ( +v
`  U )  e. 
_V
54rnex 4942 . . . 4  |-  ran  ( +v `  U )  e. 
_V
62, 3, 5fvmpt 5602 . . 3  |-  ( U  e.  _V  ->  ( BaseSet
`  U )  =  ran  ( +v `  U ) )
7 rn0 4936 . . . . 5  |-  ran  (/)  =  (/)
87eqcomi 2287 . . . 4  |-  (/)  =  ran  (/)
9 fvprc 5519 . . . 4  |-  ( -.  U  e.  _V  ->  (
BaseSet `  U )  =  (/) )
10 fvprc 5519 . . . . 5  |-  ( -.  U  e.  _V  ->  ( +v `  U )  =  (/) )
1110rneqd 4906 . . . 4  |-  ( -.  U  e.  _V  ->  ran  ( +v `  U
)  =  ran  (/) )
128, 9, 113eqtr4a 2341 . . 3  |-  ( -.  U  e.  _V  ->  (
BaseSet `  U )  =  ran  ( +v `  U ) )
136, 12pm2.61i 156 . 2  |-  ( BaseSet `  U )  =  ran  ( +v `  U )
14 bafval.1 . 2  |-  X  =  ( BaseSet `  U )
15 bafval.2 . . 3  |-  G  =  ( +v `  U
)
1615rneqi 4905 . 2  |-  ran  G  =  ran  ( +v `  U )
1713, 14, 163eqtr4i 2313 1  |-  X  =  ran  G
Colors of variables: wff set class
Syntax hints:   -. wn 3    = wceq 1623    e. wcel 1684   _Vcvv 2788   (/)c0 3455   ran crn 4690   ` cfv 5255   +vcpv 21141   BaseSetcba 21142
This theorem is referenced by:  nvi  21170  nvgf  21174  nvsf  21175  nvgcl  21176  nvcom  21177  nvass  21178  nvadd32  21180  nvrcan  21181  nvlcan  21182  nvadd4  21183  nvscl  21184  nvsid  21185  nvsass  21186  nvdi  21188  nvdir  21189  nv2  21190  nvzcl  21192  nv0rid  21193  nv0lid  21194  nv0  21195  nvsz  21196  nvinv  21197  nvinvfval  21198  nvmval  21200  nvmfval  21202  nvnnncan1  21206  nvnnncan2  21207  nvnegneg  21209  nvrinv  21211  nvlinv  21212  nvaddsubass  21216  nvaddsub  21217  nvdm  21227  nvmtri2  21238  cnnvba  21247  sspba  21303  isph  21400  phpar  21402  ip0i  21403  ipdirilem  21407  hhba  21746  hhssabloi  21839  hhshsslem1  21844
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-iota 5219  df-fun 5257  df-fv 5263  df-ba 21152
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