Theorem 00lsp 18981

Description: fvco4i 6276 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.

Step	Hyp	Ref	Expression
1		0ex 4790	. . 3 |- (/) e. _V
2		base0 15912	. . . 4 |- (/) = ( Base ` (/) )
3		00lss 18942	. . . 4 |- (/) = ( LSubSp ` (/) )
4		eqid 2622	. . . 4 |- ( LSpan ` (/) ) = ( LSpan ` (/) )
5	2, 3, 4	lspfval 18973	. . 3 |- ( (/) e. _V -> ( LSpan ` (/) ) = ( a e. ~P (/) |-> |^| { b e. (/) | a C_ b } ) )
6	1, 5	ax-mp 5	. 2 |- ( LSpan ` (/) ) = ( a e. ~P (/) |-> |^| { b e. (/) | a C_ b } )
7		eqid 2622	. . . . 5 |- ( a e. ~P (/) |-> |^| { b e. (/) | a C_ b } ) = ( a e. ~P (/) |-> |^| { b e. (/) | a C_ b } )
8	7	dmmpt 5630	. . . 4 |- dom ( a e. ~P (/) |-> |^| { b e. (/) | a C_ b } ) = { a e. ~P (/) | |^| { b e. (/) | a C_ b } e. _V }
9		vprc 4796	. . . . . . 7 |- -. _V e. _V
10		rab0 3955	. . . . . . . . . 10 |- { b e. (/) | a C_ b } = (/)
11	10	inteqi 4479	. . . . . . . . 9 |- |^| { b e. (/) | a C_ b } = |^| (/)
12		int0 4490	. . . . . . . . 9 |- |^| (/) = _V
13	11, 12	eqtri 2644	. . . . . . . 8 |- |^| { b e. (/) | a C_ b } = _V
14	13	eleq1i 2692	. . . . . . 7 |- ( |^| { b e. (/) | a C_ b } e. _V <-> _V e. _V )
15	9, 14	mtbir 313	. . . . . 6 |- -. |^| { b e. (/) | a C_ b } e. _V
16	15	rgenw 2924	. . . . 5 |- A. a e. ~P (/) -. |^| { b e. (/) | a C_ b } e. _V
17		rabeq0 3957	. . . . 5 |- ( { a e. ~P (/) | |^| { b e. (/) | a C_ b } e. _V } = (/) <-> A. a e. ~P (/) -. |^| { b e. (/) | a C_ b } e. _V )
18	16, 17	mpbir 221	. . . 4 |- { a e. ~P (/) | |^| { b e. (/) | a C_ b } e. _V } = (/)
19	8, 18	eqtri 2644	. . 3 |- dom ( a e. ~P (/) |-> |^| { b e. (/) | a C_ b } ) = (/)
20		funmpt 5926	. . . . 5 |- Fun ( a e. ~P (/) |-> |^| { b e. (/) | a C_ b } )
21		funrel 5905	. . . . 5 |- ( Fun ( a e. ~P (/) |-> |^| { b e. (/) | a C_ b } ) -> Rel ( a e. ~P (/) |-> |^| { b e. (/) | a C_ b } ) )
22	20, 21	ax-mp 5	. . . 4 |- Rel ( a e. ~P (/) |-> |^| { b e. (/) | a C_ b } )
23		reldm0 5343	. . . 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 } ) = (/) ) )
24	22, 23	ax-mp 5	. . 3 |- ( ( a e. ~P (/) |-> |^| { b e. (/) | a C_ b } ) = (/) <-> dom ( a e. ~P (/) |-> |^| { b e. (/) | a C_ b } ) = (/) )
25	19, 24	mpbir 221	. 2 |- ( a e. ~P (/) |-> |^| { b e. (/) | a C_ b } ) = (/)
26	6, 25	eqtr2i 2645	1 |- (/) = ( LSpan ` (/) )

Syntax hints:   -. wn 3   <-> wb 196   = wceq 1483   e. wcel 1990   A.wral 2912   {crab 2916   _Vcvv 3200   C_ wss 3574   (/)c0 3915   ~Pcpw 4158   |^|cint 4475   |-> cmpt 4729   dom cdm 5114   Rel wrel 5119   Fun wfun 5882   ` cfv 5888   LSpanclspn 18971

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-slot 15861  df-base 15863  df-lss 18933  df-lsp 18972

This theorem is referenced by:  rspval  19193