Theorem topfneec2 32351

Description: A topology is precisely identified with its equivalence class. (Contributed by Jeff Hankins, 12-Oct-2009.)

Hypothesis

Ref	Expression
topfneec2.1	|- .~ = ( Fne i^i `' Fne )

Assertion

Ref	Expression
topfneec2	|- ( ( J e. Top /\ K e. Top ) -> ( [ J ] .~ = [ K ] .~ <-> J = K ) )

Proof of Theorem topfneec2

Step	Hyp	Ref	Expression
1		topfneec2.1	. . 3 |- .~ = ( Fne i^i `' Fne )
2	1	fneval 32347	. 2 |- ( ( J e. Top /\ K e. Top ) -> ( J .~ K <-> ( topGen ` J ) = ( topGen ` K ) ) )
3	1	fneer 32348	. . . 4 |- .~ Er _V
4	3	a1i 11	. . 3 |- ( ( J e. Top /\ K e. Top ) -> .~ Er _V )
5		elex 3212	. . . 4 |- ( J e. Top -> J e. _V )
6	5	adantr 481	. . 3 |- ( ( J e. Top /\ K e. Top ) -> J e. _V )
7	4, 6	erth 7791	. 2 |- ( ( J e. Top /\ K e. Top ) -> ( J .~ K <-> [ J ] .~ = [ K ] .~ ) )
8		tgtop 20777	. . 3 |- ( J e. Top -> ( topGen ` J ) = J )
9		tgtop 20777	. . 3 |- ( K e. Top -> ( topGen ` K ) = K )
10	8, 9	eqeqan12d 2638	. 2 |- ( ( J e. Top /\ K e. Top ) -> ( ( topGen ` J ) = ( topGen ` K ) <-> J = K ) )
11	2, 7, 10	3bitr3d 298	1 |- ( ( J e. Top /\ K e. Top ) -> ( [ J ] .~ = [ K ] .~ <-> J = K ) )

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   ` cfv 5888   Er wer 7739   [cec 7740   topGenctg 16098   Topctop 20698   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-fne 32332

This theorem is referenced by: (None)