MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  00lsp Unicode version

Theorem 00lsp 15738
Description: fvco4i 5597 lemma for linear spans. (Contributed by Stefan O'Rear, 4-Apr-2015.)
Assertion
Ref Expression
00lsp  |-  (/)  =  (
LSpan `  (/) )

Proof of Theorem 00lsp
Dummy variables  a 
b are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 0ex 4150 . . 3  |-  (/)  e.  _V
2 base0 13185 . . . 4  |-  (/)  =  (
Base `  (/) )
3 00lss 15699 . . . 4  |-  (/)  =  (
LSubSp `  (/) )
4 eqid 2283 . . . 4  |-  ( LSpan `  (/) )  =  ( LSpan `  (/) )
52, 3, 4lspfval 15730 . . 3  |-  ( (/)  e.  _V  ->  ( LSpan `  (/) )  =  (
a  e.  ~P (/)  |->  |^| { b  e.  (/)  |  a  C_  b } ) )
61, 5ax-mp 8 . 2  |-  ( LSpan `  (/) )  =  (
a  e.  ~P (/)  |->  |^| { b  e.  (/)  |  a  C_  b } )
7 eqid 2283 . . . . 5  |-  ( a  e.  ~P (/)  |->  |^| { b  e.  (/)  |  a  C_  b } )  =  ( a  e.  ~P (/)  |->  |^| { b  e.  (/)  |  a  C_  b } )
87dmmpt 5168 . . . 4  |-  dom  (
a  e.  ~P (/)  |->  |^| { b  e.  (/)  |  a  C_  b } )  =  {
a  e.  ~P (/)  |  |^| { b  e.  (/)  |  a 
C_  b }  e.  _V }
9 vprc 4152 . . . . . . 7  |-  -.  _V  e.  _V
10 rab0 3475 . . . . . . . . . 10  |-  { b  e.  (/)  |  a  C_  b }  =  (/)
1110inteqi 3866 . . . . . . . . 9  |-  |^| { b  e.  (/)  |  a  C_  b }  =  |^| (/)
12 int0 3876 . . . . . . . . 9  |-  |^| (/)  =  _V
1311, 12eqtri 2303 . . . . . . . 8  |-  |^| { b  e.  (/)  |  a  C_  b }  =  _V
1413eleq1i 2346 . . . . . . 7  |-  ( |^| { b  e.  (/)  |  a 
C_  b }  e.  _V 
<->  _V  e.  _V )
159, 14mtbir 290 . . . . . 6  |-  -.  |^| { b  e.  (/)  |  a 
C_  b }  e.  _V
1615rgenw 2610 . . . . 5  |-  A. a  e.  ~P  (/)  -.  |^| { b  e.  (/)  |  a  C_  b }  e.  _V
17 rabeq0 3476 . . . . 5  |-  ( { a  e.  ~P (/)  |  |^| { b  e.  (/)  |  a 
C_  b }  e.  _V }  =  (/)  <->  A. a  e.  ~P  (/)  -.  |^| { b  e.  (/)  |  a  C_  b }  e.  _V )
1816, 17mpbir 200 . . . 4  |-  { a  e.  ~P (/)  |  |^| { b  e.  (/)  |  a 
C_  b }  e.  _V }  =  (/)
198, 18eqtri 2303 . . 3  |-  dom  (
a  e.  ~P (/)  |->  |^| { b  e.  (/)  |  a  C_  b } )  =  (/)
20 funmpt 5290 . . . . 5  |-  Fun  (
a  e.  ~P (/)  |->  |^| { b  e.  (/)  |  a  C_  b } )
21 funrel 5272 . . . . 5  |-  ( Fun  ( a  e.  ~P (/)  |-> 
|^| { b  e.  (/)  |  a  C_  b }
)  ->  Rel  ( a  e.  ~P (/)  |->  |^| { b  e.  (/)  |  a  C_  b } ) )
2220, 21ax-mp 8 . . . 4  |-  Rel  (
a  e.  ~P (/)  |->  |^| { b  e.  (/)  |  a  C_  b } )
23 reldm0 4896 . . . 4  |-  ( Rel  ( a  e.  ~P (/)  |-> 
|^| { b  e.  (/)  |  a  C_  b }
)  ->  ( (
a  e.  ~P (/)  |->  |^| { b  e.  (/)  |  a  C_  b } )  =  (/)  <->  dom  ( a  e.  ~P (/)  |-> 
|^| { b  e.  (/)  |  a  C_  b }
)  =  (/) ) )
2422, 23ax-mp 8 . . 3  |-  ( ( a  e.  ~P (/)  |->  |^| { b  e.  (/)  |  a  C_  b } )  =  (/)  <->  dom  ( a  e.  ~P (/)  |-> 
|^| { b  e.  (/)  |  a  C_  b }
)  =  (/) )
2519, 24mpbir 200 . 2  |-  ( a  e.  ~P (/)  |->  |^| { b  e.  (/)  |  a  C_  b } )  =  (/)
266, 25eqtr2i 2304 1  |-  (/)  =  (
LSpan `  (/) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    <-> wb 176    = wceq 1623    e. wcel 1684   A.wral 2543   {crab 2547   _Vcvv 2788    C_ wss 3152   (/)c0 3455   ~Pcpw 3625   |^|cint 3862    e. cmpt 4077   dom cdm 4689   Rel wrel 4694   Fun wfun 5249   ` cfv 5255   LSpanclspn 15728
This theorem is referenced by:  rspval  15947
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-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214
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-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  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-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-slot 13152  df-base 13153  df-lss 15690  df-lsp 15729
  Copyright terms: Public domain W3C validator