Theorem fneer 37233

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 6885	. 2 ⊢ (𝑥 = 𝑦 → (topGen‘𝑥) = (topGen‘𝑦))
2		fneval.1	. . . . . 6 ⊢ ∼ = (Fne ∩ ◡Fne)
3		inss1 4133	. . . . . 6 ⊢ (Fne ∩ ◡Fne) ⊆ Fne
4	2, 3	eqsstri 3954	. . . . 5 ⊢ ∼ ⊆ Fne
5		fnerel 37218	. . . . 5 ⊢ Rel Fne
6		relss 5757	. . . . 5 ⊢ ( ∼ ⊆ Fne → (Rel Fne → Rel ∼ ))
7	4, 5, 6	mp2 9	. . . 4 ⊢ Rel ∼
8		dfrel4v 6181	. . . 4 ⊢ (Rel ∼ ↔ ∼ = {⟨𝑥, 𝑦⟩ ∣ 𝑥 ∼ 𝑦})
9	7, 8	mpbi 233	. . 3 ⊢ ∼ = {⟨𝑥, 𝑦⟩ ∣ 𝑥 ∼ 𝑦}
10	2	fneval 37232	. . . . 5 ⊢ ((𝑥 ∈ V ∧ 𝑦 ∈ V) → (𝑥 ∼ 𝑦 ↔ (topGen‘𝑥) = (topGen‘𝑦)))
11	10	el2v 3457	. . . 4 ⊢ (𝑥 ∼ 𝑦 ↔ (topGen‘𝑥) = (topGen‘𝑦))
12	11	opabbii 5171	. . 3 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝑥 ∼ 𝑦} = {⟨𝑥, 𝑦⟩ ∣ (topGen‘𝑥) = (topGen‘𝑦)}
13	9, 12	eqtri 2783	. 2 ⊢ ∼ = {⟨𝑥, 𝑦⟩ ∣ (topGen‘𝑥) = (topGen‘𝑦)}
14	1, 13	eqer 8757	1 ⊢ ∼ Er V

This proof depends on syntax axioms:   ↔ wb 209   = wceq 1569  Vcvv 3450   ⊆ wss 3895   ∩ cin 4089   class class class wbr 5102  {copab 5166  ^◡ccnv 5649  Rel wrel 5655  ‘cfv 6538   Er wer 8717  topGenctg 17613  Fnecfne 37216

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-iota 6494  df-fun 6540  df-fv 6546  df-er 8720  df-topgen 17619  df-fne 37217

This theorem is used by:  topfneec  37235  topfneec2  37236