Theorem topfneec 32350

Description: A cover is equivalent to a topology iff it is a base for that topology. (Contributed by Jeff Hankins, 8-Oct-2009.) (Proof shortened by Mario Carneiro, 11-Sep-2015.)

Hypothesis

Ref	Expression
topfneec.1	|- .~ = ( Fne i^i `' Fne )

Assertion

Ref	Expression
topfneec	|- ( J e. Top -> ( A e. [ J ] .~ <-> ( topGen ` A ) = J ) )

Proof of Theorem topfneec

Step	Hyp	Ref	Expression
1		topfneec.1	. . . . 5 |- .~ = ( Fne i^i `' Fne )
2	1	fneer 32348	. . . 4 |- .~ Er _V
3		errel 7751	. . . 4 |- ( .~ Er _V -> Rel .~ )
4	2, 3	ax-mp 5	. . 3 |- Rel .~
5		relelec 7787	. . 3 |- ( Rel .~ -> ( A e. [ J ] .~ <-> J .~ A ) )
6	4, 5	ax-mp 5	. 2 |- ( A e. [ J ] .~ <-> J .~ A )
7	4	brrelex2i 5159	. . . 4 |- ( J .~ A -> A e. _V )
8	7	a1i 11	. . 3 |- ( J e. Top -> ( J .~ A -> A e. _V ) )
9		eleq1 2689	. . . . . . 7 |- ( ( topGen ` A ) = J -> ( ( topGen ` A ) e. Top <-> J e. Top ) )
10	9	biimparc 504	. . . . . 6 |- ( ( J e. Top /\ ( topGen ` A ) = J ) -> ( topGen ` A ) e. Top )
11		tgclb 20774	. . . . . 6 |- ( A e. TopBases <-> ( topGen ` A ) e. Top )
12	10, 11	sylibr 224	. . . . 5 |- ( ( J e. Top /\ ( topGen ` A ) = J ) -> A e. TopBases )
13		elex 3212	. . . . 5 |- ( A e. TopBases -> A e. _V )
14	12, 13	syl 17	. . . 4 |- ( ( J e. Top /\ ( topGen ` A ) = J ) -> A e. _V )
15	14	ex 450	. . 3 |- ( J e. Top -> ( ( topGen ` A ) = J -> A e. _V ) )
16	1	fneval 32347	. . . . 5 |- ( ( J e. Top /\ A e. _V ) -> ( J .~ A <-> ( topGen ` J ) = ( topGen ` A ) ) )
17		tgtop 20777	. . . . . . . 8 |- ( J e. Top -> ( topGen ` J ) = J )
18	17	eqeq1d 2624	. . . . . . 7 |- ( J e. Top -> ( ( topGen ` J ) = ( topGen ` A ) <-> J = ( topGen ` A ) ) )
19		eqcom 2629	. . . . . . 7 |- ( J = ( topGen ` A ) <-> ( topGen ` A ) = J )
20	18, 19	syl6bb 276	. . . . . 6 |- ( J e. Top -> ( ( topGen ` J ) = ( topGen ` A ) <-> ( topGen ` A ) = J ) )
21	20	adantr 481	. . . . 5 |- ( ( J e. Top /\ A e. _V ) -> ( ( topGen ` J ) = ( topGen ` A ) <-> ( topGen ` A ) = J ) )
22	16, 21	bitrd 268	. . . 4 |- ( ( J e. Top /\ A e. _V ) -> ( J .~ A <-> ( topGen ` A ) = J ) )
23	22	ex 450	. . 3 |- ( J e. Top -> ( A e. _V -> ( J .~ A <-> ( topGen ` A ) = J ) ) )
24	8, 15, 23	pm5.21ndd 369	. 2 |- ( J e. Top -> ( J .~ A <-> ( topGen ` A ) = J ) )
25	6, 24	syl5bb 272	1 |- ( J e. Top -> ( A e. [ J ] .~ <-> ( topGen ` A ) = J ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   i^i cin 3573   class class class wbr 4653   `'ccnv 5113   Rel wrel 5119   ` cfv 5888   Er wer 7739   [cec 7740   topGenctg 16098   Topctop 20698   TopBasesctb 20749   Fnecfne 32331

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-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

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-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-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-fv 5896  df-er 7742  df-ec 7744  df-topgen 16104  df-top 20699  df-bases 20750  df-fne 32332

This theorem is referenced by: (None)