Theorem fneer 34469

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 ∩ ◡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 6756	. 2 ⊢ (𝑥 = 𝑦 → (topGen‘𝑥) = (topGen‘𝑦))
2		fneval.1	. . . . . 6 ⊢ ∼ = (Fne ∩ ◡Fne)
3		inss1 4159	. . . . . 6 ⊢ (Fne ∩ ◡Fne) ⊆ Fne
4	2, 3	eqsstri 3951	. . . . 5 ⊢ ∼ ⊆ Fne
5		fnerel 34454	. . . . 5 ⊢ Rel Fne
6		relss 5682	. . . . 5 ⊢ ( ∼ ⊆ Fne → (Rel Fne → Rel ∼ ))
7	4, 5, 6	mp2 9	. . . 4 ⊢ Rel ∼
8		dfrel4v 6082	. . . 4 ⊢ (Rel ∼ ↔ ∼ = {⟨𝑥, 𝑦⟩ ∣ 𝑥 ∼ 𝑦})
9	7, 8	mpbi 229	. . 3 ⊢ ∼ = {⟨𝑥, 𝑦⟩ ∣ 𝑥 ∼ 𝑦}
10	2	fneval 34468	. . . . 5 ⊢ ((𝑥 ∈ V ∧ 𝑦 ∈ V) → (𝑥 ∼ 𝑦 ↔ (topGen‘𝑥) = (topGen‘𝑦)))
11	10	el2v 3430	. . . 4 ⊢ (𝑥 ∼ 𝑦 ↔ (topGen‘𝑥) = (topGen‘𝑦))
12	11	opabbii 5137	. . 3 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝑥 ∼ 𝑦} = {⟨𝑥, 𝑦⟩ ∣ (topGen‘𝑥) = (topGen‘𝑦)}
13	9, 12	eqtri 2766	. 2 ⊢ ∼ = {⟨𝑥, 𝑦⟩ ∣ (topGen‘𝑥) = (topGen‘𝑦)}
14	1, 13	eqer 8491	1 ⊢ ∼ Er V

Syntax hints:   ↔ wb 205   = wceq 1539  Vcvv 3422   ∩ cin 3882   ⊆ wss 3883   class class class wbr 5070  {copab 5132  ^◡ccnv 5579  Rel wrel 5585  ‘cfv 6418   Er wer 8453  topGenctg 17065  Fnecfne 34452

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-iota 6376  df-fun 6420  df-fv 6426  df-er 8456  df-topgen 17071  df-fne 34453

This theorem is referenced by:  topfneec  34471  topfneec2  34472