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Theorem 00lsp 15754
Description: fvco4i 5613 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 4166 . . 3  |-  (/)  e.  _V
2 base0 13201 . . . 4  |-  (/)  =  (
Base `  (/) )
3 00lss 15715 . . . 4  |-  (/)  =  (
LSubSp `  (/) )
4 eqid 2296 . . . 4  |-  ( LSpan `  (/) )  =  ( LSpan `  (/) )
52, 3, 4lspfval 15746 . . 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 2296 . . . . 5  |-  ( a  e.  ~P (/)  |->  |^| { b  e.  (/)  |  a  C_  b } )  =  ( a  e.  ~P (/)  |->  |^| { b  e.  (/)  |  a  C_  b } )
87dmmpt 5184 . . . 4  |-  dom  (
a  e.  ~P (/)  |->  |^| { b  e.  (/)  |  a  C_  b } )  =  {
a  e.  ~P (/)  |  |^| { b  e.  (/)  |  a 
C_  b }  e.  _V }
9 vprc 4168 . . . . . . 7  |-  -.  _V  e.  _V
10 rab0 3488 . . . . . . . . . 10  |-  { b  e.  (/)  |  a  C_  b }  =  (/)
1110inteqi 3882 . . . . . . . . 9  |-  |^| { b  e.  (/)  |  a  C_  b }  =  |^| (/)
12 int0 3892 . . . . . . . . 9  |-  |^| (/)  =  _V
1311, 12eqtri 2316 . . . . . . . 8  |-  |^| { b  e.  (/)  |  a  C_  b }  =  _V
1413eleq1i 2359 . . . . . . 7  |-  ( |^| { b  e.  (/)  |  a 
C_  b }  e.  _V 
<->  _V  e.  _V )
159, 14mtbir 290 . . . . . 6  |-  -.  |^| { b  e.  (/)  |  a 
C_  b }  e.  _V
1615rgenw 2623 . . . . 5  |-  A. a  e.  ~P  (/)  -.  |^| { b  e.  (/)  |  a  C_  b }  e.  _V
17 rabeq0 3489 . . . . 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 2316 . . 3  |-  dom  (
a  e.  ~P (/)  |->  |^| { b  e.  (/)  |  a  C_  b } )  =  (/)
20 funmpt 5306 . . . . 5  |-  Fun  (
a  e.  ~P (/)  |->  |^| { b  e.  (/)  |  a  C_  b } )
21 funrel 5288 . . . . 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 4912 . . . 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 2317 1  |-  (/)  =  (
LSpan `  (/) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    <-> wb 176    = wceq 1632    e. wcel 1696   A.wral 2556   {crab 2560   _Vcvv 2801    C_ wss 3165   (/)c0 3468   ~Pcpw 3638   |^|cint 3878    e. cmpt 4093   dom cdm 4705   Rel wrel 4710   Fun wfun 5265   ` cfv 5271   LSpanclspn 15744
This theorem is referenced by:  rspval  15963
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-slot 13168  df-base 13169  df-lss 15706  df-lsp 15745
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