Theorem fneer 36726

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 6871	. 2 ⊢ (𝑥 = 𝑦 → (topGen‘𝑥) = (topGen‘𝑦))
2		fneval.1	. . . . . 6 ⊢ ∼ = (Fne ∩ ◡Fne)
3		inss1 4191	. . . . . 6 ⊢ (Fne ∩ ◡Fne) ⊆ Fne
4	2, 3	eqsstri 3985	. . . . 5 ⊢ ∼ ⊆ Fne
5		fnerel 36711	. . . . 5 ⊢ Rel Fne
6		relss 5759	. . . . 5 ⊢ ( ∼ ⊆ Fne → (Rel Fne → Rel ∼ ))
7	4, 5, 6	mp2 9	. . . 4 ⊢ Rel ∼
8		dfrel4v 6180	. . . 4 ⊢ (Rel ∼ ↔ ∼ = {⟨𝑥, 𝑦⟩ ∣ 𝑥 ∼ 𝑦})
9	7, 8	mpbi 233	. . 3 ⊢ ∼ = {⟨𝑥, 𝑦⟩ ∣ 𝑥 ∼ 𝑦}
10	2	fneval 36725	. . . . 5 ⊢ ((𝑥 ∈ V ∧ 𝑦 ∈ V) → (𝑥 ∼ 𝑦 ↔ (topGen‘𝑥) = (topGen‘𝑦)))
11	10	el2v 3464	. . . 4 ⊢ (𝑥 ∼ 𝑦 ↔ (topGen‘𝑥) = (topGen‘𝑦))
12	11	opabbii 5172	. . 3 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝑥 ∼ 𝑦} = {⟨𝑥, 𝑦⟩ ∣ (topGen‘𝑥) = (topGen‘𝑦)}
13	9, 12	eqtri 2788	. 2 ⊢ ∼ = {⟨𝑥, 𝑦⟩ ∣ (topGen‘𝑥) = (topGen‘𝑦)}
14	1, 13	eqer 8719	1 ⊢ ∼ Er V

Syntax hints:   ↔ wb 209   = wceq 1563  Vcvv 3457   ∩ cin 3906   ⊆ wss 3907   class class class wbr 5105  {copab 5167  ^◡ccnv 5651  Rel wrel 5657  ‘cfv 6525   Er wer 8679  topGenctg 17480  Fnecfne 36709

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-sep 5251  ax-nul 5261  ax-pow 5327  ax-pr 5395  ax-un 7722

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3080  df-rex 3090  df-rab 3418  df-v 3459  df-sbc 3748  df-csb 3856  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-pw 4560  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4869  df-iun 4954  df-br 5106  df-opab 5168  df-mpt 5187  df-id 5547  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-iota 6481  df-fun 6527  df-fv 6533  df-er 8682  df-topgen 17486  df-fne 36710

This theorem is referenced by:  topfneec  36728  topfneec2  36729