Theorem fneer 33701

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 6670	. 2 ⊢ (𝑥 = 𝑦 → (topGen‘𝑥) = (topGen‘𝑦))
2		fneval.1	. . . . . 6 ⊢ ∼ = (Fne ∩ ◡Fne)
3		inss1 4205	. . . . . 6 ⊢ (Fne ∩ ◡Fne) ⊆ Fne
4	2, 3	eqsstri 4001	. . . . 5 ⊢ ∼ ⊆ Fne
5		fnerel 33686	. . . . 5 ⊢ Rel Fne
6		relss 5656	. . . . 5 ⊢ ( ∼ ⊆ Fne → (Rel Fne → Rel ∼ ))
7	4, 5, 6	mp2 9	. . . 4 ⊢ Rel ∼
8		dfrel4v 6047	. . . 4 ⊢ (Rel ∼ ↔ ∼ = {⟨𝑥, 𝑦⟩ ∣ 𝑥 ∼ 𝑦})
9	7, 8	mpbi 232	. . 3 ⊢ ∼ = {⟨𝑥, 𝑦⟩ ∣ 𝑥 ∼ 𝑦}
10	2	fneval 33700	. . . . 5 ⊢ ((𝑥 ∈ V ∧ 𝑦 ∈ V) → (𝑥 ∼ 𝑦 ↔ (topGen‘𝑥) = (topGen‘𝑦)))
11	10	el2v 3501	. . . 4 ⊢ (𝑥 ∼ 𝑦 ↔ (topGen‘𝑥) = (topGen‘𝑦))
12	11	opabbii 5133	. . 3 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝑥 ∼ 𝑦} = {⟨𝑥, 𝑦⟩ ∣ (topGen‘𝑥) = (topGen‘𝑦)}
13	9, 12	eqtri 2844	. 2 ⊢ ∼ = {⟨𝑥, 𝑦⟩ ∣ (topGen‘𝑥) = (topGen‘𝑦)}
14	1, 13	eqer 8324	1 ⊢ ∼ Er V

Syntax hints:   ↔ wb 208   = wceq 1537  Vcvv 3494   ∩ cin 3935   ⊆ wss 3936   class class class wbr 5066  {copab 5128  ^◡ccnv 5554  Rel wrel 5560  ‘cfv 6355   Er wer 8286  topGenctg 16711  Fnecfne 33684

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-sep 5203  ax-nul 5210  ax-pow 5266  ax-pr 5330  ax-un 7461

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4839  df-iun 4921  df-br 5067  df-opab 5129  df-mpt 5147  df-id 5460  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-iota 6314  df-fun 6357  df-fv 6363  df-er 8289  df-topgen 16717  df-fne 33685

This theorem is referenced by:  topfneec  33703  topfneec2  33704