Theorem fneer 32348

Description: Fineness intersected with its converse is an equivalence relation. (Contributed by Jeff Hankins, 6-Oct-2009.) (Revised by Mario Carneiro, 11-Sep-2015.)

Hypothesis

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

Assertion

Ref	Expression
fneer	|- .~ Er _V

Proof of Theorem fneer

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 6191	. 2 |- ( x = y -> ( topGen ` x ) = ( topGen ` y ) )
2		fneval.1	. . . . . 6 |- .~ = ( Fne i^i `' Fne )
3		inss1 3833	. . . . . 6 |- ( Fne i^i `' Fne ) C_ Fne
4	2, 3	eqsstri 3635	. . . . 5 |- .~ C_ Fne
5		fnerel 32333	. . . . 5 |- Rel Fne
6		relss 5206	. . . . 5 |- ( .~ C_ Fne -> ( Rel Fne -> Rel .~ ) )
7	4, 5, 6	mp2 9	. . . 4 |- Rel .~
8		dfrel4v 5584	. . . 4 |- ( Rel .~ <-> .~ = { <. x , y >. | x .~ y } )
9	7, 8	mpbi 220	. . 3 |- .~ = { <. x , y >. | x .~ y }
10		vex 3203	. . . . 5 |- x e. _V
11		vex 3203	. . . . 5 |- y e. _V
12	2	fneval 32347	. . . . 5 |- ( ( x e. _V /\ y e. _V ) -> ( x .~ y <-> ( topGen ` x ) = ( topGen ` y ) ) )
13	10, 11, 12	mp2an 708	. . . 4 |- ( x .~ y <-> ( topGen ` x ) = ( topGen ` y ) )
14	13	opabbii 4717	. . 3 |- { <. x , y >. | x .~ y } = { <. x , y >. | ( topGen ` x ) = ( topGen ` y ) }
15	9, 14	eqtri 2644	. 2 |- .~ = { <. x , y >. | ( topGen ` x ) = ( topGen ` y ) }
16	1, 15	eqer 7777	1 |- .~ Er _V

Syntax hints:   <-> wb 196   = wceq 1483   e. wcel 1990   _Vcvv 3200   i^i cin 3573   C_ wss 3574   class class class wbr 4653   {copab 4712   `'ccnv 5113   Rel wrel 5119   ` cfv 5888   Er wer 7739   topGenctg 16098   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-iota 5851  df-fun 5890  df-fv 5896  df-er 7742  df-topgen 16104  df-fne 32332

This theorem is referenced by:  topfneec  32350  topfneec2  32351