Theorem islinds 20148

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	|- B = ( Base ` W )

Assertion

Ref	Expression
islinds	|- ( W e. V -> ( X e. (LIndS ` W ) <-> ( X C_ B /\ ( _I |` X ) LIndF W ) ) )

Proof of Theorem islinds

Dummy variables s w are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 3212	. . . . 5 |- ( W e. V -> W e. _V )
2		fveq2 6191	. . . . . . . 8 |- ( w = W -> ( Base ` w ) = ( Base ` W ) )
3	2	pweqd 4163	. . . . . . 7 |- ( w = W -> ~P ( Base ` w ) = ~P ( Base ` W ) )
4		breq2 4657	. . . . . . 7 |- ( w = W -> ( ( _I |` s ) LIndF w <-> ( _I |` s ) LIndF W ) )
5	3, 4	rabeqbidv 3195	. . . . . 6 |- ( w = W -> { s e. ~P ( Base ` w ) | ( _I |` s ) LIndF w } = { s e. ~P ( Base ` W ) | ( _I |` s ) LIndF W } )
6		df-linds 20146	. . . . . 6 |- LIndS = ( w e. _V |-> { s e. ~P ( Base ` w ) | ( _I |` s ) LIndF w } )
7		fvex 6201	. . . . . . . 8 |- ( Base ` W ) e. _V
8	7	pwex 4848	. . . . . . 7 |- ~P ( Base ` W ) e. _V
9	8	rabex 4813	. . . . . 6 |- { s e. ~P ( Base ` W ) | ( _I |` s ) LIndF W } e. _V
10	5, 6, 9	fvmpt 6282	. . . . 5 |- ( W e. _V -> (LIndS ` W ) = { s e. ~P ( Base ` W ) | ( _I |` s ) LIndF W } )
11	1, 10	syl 17	. . . 4 |- ( W e. V -> (LIndS ` W ) = { s e. ~P ( Base ` W ) | ( _I |` s ) LIndF W } )
12	11	eleq2d 2687	. . 3 |- ( W e. V -> ( X e. (LIndS ` W ) <-> X e. { s e. ~P ( Base ` W ) | ( _I |` s ) LIndF W } ) )
13		reseq2 5391	. . . . 5 |- ( s = X -> ( _I |` s ) = ( _I |` X ) )
14	13	breq1d 4663	. . . 4 |- ( s = X -> ( ( _I |` s ) LIndF W <-> ( _I |` X ) LIndF W ) )
15	14	elrab 3363	. . 3 |- ( X e. { s e. ~P ( Base ` W ) | ( _I |` s ) LIndF W } <-> ( X e. ~P ( Base ` W ) /\ ( _I |` X ) LIndF W ) )
16	12, 15	syl6bb 276	. 2 |- ( W e. V -> ( X e. (LIndS ` W ) <-> ( X e. ~P ( Base ` W ) /\ ( _I |` X ) LIndF W ) ) )
17	7	elpw2 4828	. . . 4 |- ( X e. ~P ( Base ` W ) <-> X C_ ( Base ` W ) )
18		islinds.b	. . . . 5 |- B = ( Base ` W )
19	18	sseq2i 3630	. . . 4 |- ( X C_ B <-> X C_ ( Base ` W ) )
20	17, 19	bitr4i 267	. . 3 |- ( X e. ~P ( Base ` W ) <-> X C_ B )
21	20	anbi1i 731	. 2 |- ( ( X e. ~P ( Base ` W ) /\ ( _I |` X ) LIndF W ) <-> ( X C_ B /\ ( _I |` X ) LIndF W ) )
22	16, 21	syl6bb 276	1 |- ( W e. V -> ( X e. (LIndS ` W ) <-> ( X C_ B /\ ( _I |` X ) LIndF W ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   {crab 2916   _Vcvv 3200   C_ wss 3574   ~Pcpw 4158   class class class wbr 4653   _I cid 5023   |` cres 5116   ` cfv 5888   Basecbs 15857   LIndF clindf 20143  LIndSclinds 20144

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-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  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-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  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-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-res 5126  df-iota 5851  df-fun 5890  df-fv 5896  df-linds 20146

This theorem is referenced by:  linds1  20149  linds2  20150  islinds2  20152  lindsss  20163  lindsmm  20167  lsslinds  20170