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Theorem islinds 27247
 Description: Property of an independent set of vectors in terms of an independent family. (Contributed by Stefan O'Rear, 24-Feb-2015.)
Hypothesis
Ref Expression
islinds.b
Assertion
Ref Expression
islinds LIndS LIndF

Proof of Theorem islinds
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elex 2956 . . . . 5
2 fveq2 5720 . . . . . . . 8
32pweqd 3796 . . . . . . 7
4 breq2 4208 . . . . . . 7 LIndF LIndF
53, 4rabeqbidv 2943 . . . . . 6 LIndF LIndF
6 df-linds 27245 . . . . . 6 LIndS LIndF
7 fvex 5734 . . . . . . . 8
87pwex 4374 . . . . . . 7
98rabex 4346 . . . . . 6 LIndF
105, 6, 9fvmpt 5798 . . . . 5 LIndS LIndF
111, 10syl 16 . . . 4 LIndS LIndF
1211eleq2d 2502 . . 3 LIndS LIndF
13 reseq2 5133 . . . . 5
1413breq1d 4214 . . . 4 LIndF LIndF
1514elrab 3084 . . 3 LIndF LIndF
1612, 15syl6bb 253 . 2 LIndS LIndF
177elpw2 4356 . . . 4
18 islinds.b . . . . 5
1918sseq2i 3365 . . . 4
2017, 19bitr4i 244 . . 3
2120anbi1i 677 . 2 LIndF LIndF
2216, 21syl6bb 253 1 LIndS LIndF
 Colors of variables: wff set class Syntax hints:   wi 4   wb 177   wa 359   wceq 1652   wcel 1725  crab 2701  cvv 2948   wss 3312  cpw 3791   class class class wbr 4204   cid 4485   cres 4872  cfv 5446  cbs 13461   LIndF clindf 27242  LIndSclinds 27243 This theorem is referenced by:  linds1  27248  linds2  27249  islinds2  27251  lindsss  27262  lindsmm  27266  lsslinds  27269 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-res 4882  df-iota 5410  df-fun 5448  df-fv 5454  df-linds 27245
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