Theorem fneer 35541

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 6891	. 2 ⊢ (𝑥 = 𝑦 → (topGen‘𝑥) = (topGen‘𝑦))
2		fneval.1	. . . . . 6 ⊢ ∼ = (Fne ∩ ◡Fne)
3		inss1 4228	. . . . . 6 ⊢ (Fne ∩ ◡Fne) ⊆ Fne
4	2, 3	eqsstri 4016	. . . . 5 ⊢ ∼ ⊆ Fne
5		fnerel 35526	. . . . 5 ⊢ Rel Fne
6		relss 5781	. . . . 5 ⊢ ( ∼ ⊆ Fne → (Rel Fne → Rel ∼ ))
7	4, 5, 6	mp2 9	. . . 4 ⊢ Rel ∼
8		dfrel4v 6189	. . . 4 ⊢ (Rel ∼ ↔ ∼ = {⟨𝑥, 𝑦⟩ ∣ 𝑥 ∼ 𝑦})
9	7, 8	mpbi 229	. . 3 ⊢ ∼ = {⟨𝑥, 𝑦⟩ ∣ 𝑥 ∼ 𝑦}
10	2	fneval 35540	. . . . 5 ⊢ ((𝑥 ∈ V ∧ 𝑦 ∈ V) → (𝑥 ∼ 𝑦 ↔ (topGen‘𝑥) = (topGen‘𝑦)))
11	10	el2v 3482	. . . 4 ⊢ (𝑥 ∼ 𝑦 ↔ (topGen‘𝑥) = (topGen‘𝑦))
12	11	opabbii 5215	. . 3 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝑥 ∼ 𝑦} = {⟨𝑥, 𝑦⟩ ∣ (topGen‘𝑥) = (topGen‘𝑦)}
13	9, 12	eqtri 2760	. 2 ⊢ ∼ = {⟨𝑥, 𝑦⟩ ∣ (topGen‘𝑥) = (topGen‘𝑦)}
14	1, 13	eqer 8740	1 ⊢ ∼ Er V

Syntax hints:   ↔ wb 205   = wceq 1541  Vcvv 3474   ∩ cin 3947   ⊆ wss 3948   class class class wbr 5148  {copab 5210  ^◡ccnv 5675  Rel wrel 5681  ‘cfv 6543   Er wer 8702  topGenctg 17387  Fnecfne 35524

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7727

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-iota 6495  df-fun 6545  df-fv 6551  df-er 8705  df-topgen 17393  df-fne 35525

This theorem is referenced by:  topfneec  35543  topfneec2  35544