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Theorem 00lsp 17410
Description: fvco4i 5943 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 4577 . . 3  |-  (/)  e.  _V
2 base0 14525 . . . 4  |-  (/)  =  (
Base `  (/) )
3 00lss 17371 . . . 4  |-  (/)  =  (
LSubSp `  (/) )
4 eqid 2467 . . . 4  |-  ( LSpan `  (/) )  =  ( LSpan `  (/) )
52, 3, 4lspfval 17402 . . 3  |-  ( (/)  e.  _V  ->  ( LSpan `  (/) )  =  (
a  e.  ~P (/)  |->  |^| { b  e.  (/)  |  a  C_  b } ) )
61, 5ax-mp 5 . 2  |-  ( LSpan `  (/) )  =  (
a  e.  ~P (/)  |->  |^| { b  e.  (/)  |  a  C_  b } )
7 eqid 2467 . . . . 5  |-  ( a  e.  ~P (/)  |->  |^| { b  e.  (/)  |  a  C_  b } )  =  ( a  e.  ~P (/)  |->  |^| { b  e.  (/)  |  a  C_  b } )
87dmmpt 5500 . . . 4  |-  dom  (
a  e.  ~P (/)  |->  |^| { b  e.  (/)  |  a  C_  b } )  =  {
a  e.  ~P (/)  |  |^| { b  e.  (/)  |  a 
C_  b }  e.  _V }
9 vprc 4585 . . . . . . 7  |-  -.  _V  e.  _V
10 rab0 3806 . . . . . . . . . 10  |-  { b  e.  (/)  |  a  C_  b }  =  (/)
1110inteqi 4286 . . . . . . . . 9  |-  |^| { b  e.  (/)  |  a  C_  b }  =  |^| (/)
12 int0 4296 . . . . . . . . 9  |-  |^| (/)  =  _V
1311, 12eqtri 2496 . . . . . . . 8  |-  |^| { b  e.  (/)  |  a  C_  b }  =  _V
1413eleq1i 2544 . . . . . . 7  |-  ( |^| { b  e.  (/)  |  a 
C_  b }  e.  _V 
<->  _V  e.  _V )
159, 14mtbir 299 . . . . . 6  |-  -.  |^| { b  e.  (/)  |  a 
C_  b }  e.  _V
1615rgenw 2825 . . . . 5  |-  A. a  e.  ~P  (/)  -.  |^| { b  e.  (/)  |  a  C_  b }  e.  _V
17 rabeq0 3807 . . . . 5  |-  ( { a  e.  ~P (/)  |  |^| { b  e.  (/)  |  a 
C_  b }  e.  _V }  =  (/)  <->  A. a  e.  ~P  (/)  -.  |^| { b  e.  (/)  |  a  C_  b }  e.  _V )
1816, 17mpbir 209 . . . 4  |-  { a  e.  ~P (/)  |  |^| { b  e.  (/)  |  a 
C_  b }  e.  _V }  =  (/)
198, 18eqtri 2496 . . 3  |-  dom  (
a  e.  ~P (/)  |->  |^| { b  e.  (/)  |  a  C_  b } )  =  (/)
20 funmpt 5622 . . . . 5  |-  Fun  (
a  e.  ~P (/)  |->  |^| { b  e.  (/)  |  a  C_  b } )
21 funrel 5603 . . . . 5  |-  ( Fun  ( a  e.  ~P (/)  |-> 
|^| { b  e.  (/)  |  a  C_  b }
)  ->  Rel  ( a  e.  ~P (/)  |->  |^| { b  e.  (/)  |  a  C_  b } ) )
2220, 21ax-mp 5 . . . 4  |-  Rel  (
a  e.  ~P (/)  |->  |^| { b  e.  (/)  |  a  C_  b } )
23 reldm0 5218 . . . 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 5 . . 3  |-  ( ( a  e.  ~P (/)  |->  |^| { b  e.  (/)  |  a  C_  b } )  =  (/)  <->  dom  ( a  e.  ~P (/)  |-> 
|^| { b  e.  (/)  |  a  C_  b }
)  =  (/) )
2519, 24mpbir 209 . 2  |-  ( a  e.  ~P (/)  |->  |^| { b  e.  (/)  |  a  C_  b } )  =  (/)
266, 25eqtr2i 2497 1  |-  (/)  =  (
LSpan `  (/) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    <-> wb 184    = wceq 1379    e. wcel 1767   A.wral 2814   {crab 2818   _Vcvv 3113    C_ wss 3476   (/)c0 3785   ~Pcpw 4010   |^|cint 4282    |-> cmpt 4505   dom cdm 4999   Rel wrel 5004   Fun wfun 5580   ` cfv 5586   LSpanclspn 17400
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-ov 6285  df-slot 14490  df-base 14491  df-lss 17362  df-lsp 17401
This theorem is referenced by:  rspval  17622
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