Theorem fneval 36318

Description: Two covers are finer than each other iff they are both bases for the same topology. (Contributed by Mario Carneiro, 11-Sep-2015.)

Hypothesis

Ref	Expression
fneval.1	⊢ ∼ = (Fne ∩ ◡Fne)

Assertion

Ref	Expression
fneval	⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ∼ 𝐵 ↔ (topGen‘𝐴) = (topGen‘𝐵)))

Proof of Theorem fneval

Step	Hyp	Ref	Expression
1		fneval.1	. . . 4 ⊢ ∼ = (Fne ∩ ◡Fne)
2	1	breqi 5172	. . 3 ⊢ (𝐴 ∼ 𝐵 ↔ 𝐴(Fne ∩ ◡Fne)𝐵)
3		brin 5218	. . . 4 ⊢ (𝐴(Fne ∩ ◡Fne)𝐵 ↔ (𝐴Fne𝐵 ∧ 𝐴◡Fne𝐵))
4		fnerel 36304	. . . . . 6 ⊢ Rel Fne
5	4	relbrcnv 6137	. . . . 5 ⊢ (𝐴◡Fne𝐵 ↔ 𝐵Fne𝐴)
6	5	anbi2i 622	. . . 4 ⊢ ((𝐴Fne𝐵 ∧ 𝐴◡Fne𝐵) ↔ (𝐴Fne𝐵 ∧ 𝐵Fne𝐴))
7	3, 6	bitri 275	. . 3 ⊢ (𝐴(Fne ∩ ◡Fne)𝐵 ↔ (𝐴Fne𝐵 ∧ 𝐵Fne𝐴))
8	2, 7	bitri 275	. 2 ⊢ (𝐴 ∼ 𝐵 ↔ (𝐴Fne𝐵 ∧ 𝐵Fne𝐴))
9		eqid 2740	. . . . . 6 ⊢ ∪ 𝐴 = ∪ 𝐴
10		eqid 2740	. . . . . 6 ⊢ ∪ 𝐵 = ∪ 𝐵
11	9, 10	isfne4b 36307	. . . . 5 ⊢ (𝐵 ∈ 𝑊 → (𝐴Fne𝐵 ↔ (∪ 𝐴 = ∪ 𝐵 ∧ (topGen‘𝐴) ⊆ (topGen‘𝐵))))
12	10, 9	isfne4b 36307	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → (𝐵Fne𝐴 ↔ (∪ 𝐵 = ∪ 𝐴 ∧ (topGen‘𝐵) ⊆ (topGen‘𝐴))))
13		eqcom 2747	. . . . . . 7 ⊢ (∪ 𝐵 = ∪ 𝐴 ↔ ∪ 𝐴 = ∪ 𝐵)
14	13	anbi1i 623	. . . . . 6 ⊢ ((∪ 𝐵 = ∪ 𝐴 ∧ (topGen‘𝐵) ⊆ (topGen‘𝐴)) ↔ (∪ 𝐴 = ∪ 𝐵 ∧ (topGen‘𝐵) ⊆ (topGen‘𝐴)))
15	12, 14	bitrdi 287	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → (𝐵Fne𝐴 ↔ (∪ 𝐴 = ∪ 𝐵 ∧ (topGen‘𝐵) ⊆ (topGen‘𝐴))))
16	11, 15	bi2anan9r 638	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((𝐴Fne𝐵 ∧ 𝐵Fne𝐴) ↔ ((∪ 𝐴 = ∪ 𝐵 ∧ (topGen‘𝐴) ⊆ (topGen‘𝐵)) ∧ (∪ 𝐴 = ∪ 𝐵 ∧ (topGen‘𝐵) ⊆ (topGen‘𝐴)))))
17		eqss 4024	. . . . . 6 ⊢ ((topGen‘𝐴) = (topGen‘𝐵) ↔ ((topGen‘𝐴) ⊆ (topGen‘𝐵) ∧ (topGen‘𝐵) ⊆ (topGen‘𝐴)))
18	17	anbi2i 622	. . . . 5 ⊢ ((∪ 𝐴 = ∪ 𝐵 ∧ (topGen‘𝐴) = (topGen‘𝐵)) ↔ (∪ 𝐴 = ∪ 𝐵 ∧ ((topGen‘𝐴) ⊆ (topGen‘𝐵) ∧ (topGen‘𝐵) ⊆ (topGen‘𝐴))))
19		anandi 675	. . . . 5 ⊢ ((∪ 𝐴 = ∪ 𝐵 ∧ ((topGen‘𝐴) ⊆ (topGen‘𝐵) ∧ (topGen‘𝐵) ⊆ (topGen‘𝐴))) ↔ ((∪ 𝐴 = ∪ 𝐵 ∧ (topGen‘𝐴) ⊆ (topGen‘𝐵)) ∧ (∪ 𝐴 = ∪ 𝐵 ∧ (topGen‘𝐵) ⊆ (topGen‘𝐴))))
20	18, 19	bitri 275	. . . 4 ⊢ ((∪ 𝐴 = ∪ 𝐵 ∧ (topGen‘𝐴) = (topGen‘𝐵)) ↔ ((∪ 𝐴 = ∪ 𝐵 ∧ (topGen‘𝐴) ⊆ (topGen‘𝐵)) ∧ (∪ 𝐴 = ∪ 𝐵 ∧ (topGen‘𝐵) ⊆ (topGen‘𝐴))))
21	16, 20	bitr4di 289	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((𝐴Fne𝐵 ∧ 𝐵Fne𝐴) ↔ (∪ 𝐴 = ∪ 𝐵 ∧ (topGen‘𝐴) = (topGen‘𝐵))))
22		unieq 4942	. . . . 5 ⊢ ((topGen‘𝐴) = (topGen‘𝐵) → ∪ (topGen‘𝐴) = ∪ (topGen‘𝐵))
23		unitg 22995	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → ∪ (topGen‘𝐴) = ∪ 𝐴)
24		unitg 22995	. . . . . 6 ⊢ (𝐵 ∈ 𝑊 → ∪ (topGen‘𝐵) = ∪ 𝐵)
25	23, 24	eqeqan12d 2754	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (∪ (topGen‘𝐴) = ∪ (topGen‘𝐵) ↔ ∪ 𝐴 = ∪ 𝐵))
26	22, 25	imbitrid 244	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((topGen‘𝐴) = (topGen‘𝐵) → ∪ 𝐴 = ∪ 𝐵))
27	26	pm4.71rd 562	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((topGen‘𝐴) = (topGen‘𝐵) ↔ (∪ 𝐴 = ∪ 𝐵 ∧ (topGen‘𝐴) = (topGen‘𝐵))))
28	21, 27	bitr4d 282	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((𝐴Fne𝐵 ∧ 𝐵Fne𝐴) ↔ (topGen‘𝐴) = (topGen‘𝐵)))
29	8, 28	bitrid 283	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ∼ 𝐵 ↔ (topGen‘𝐴) = (topGen‘𝐵)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1537   ∈ wcel 2108   ∩ cin 3975   ⊆ wss 3976  ∪ cuni 4931   class class class wbr 5166  ^◡ccnv 5699  ‘cfv 6573  topGenctg 17497  Fnecfne 36302

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-iota 6525  df-fun 6575  df-fv 6581  df-topgen 17503  df-fne 36303

This theorem is referenced by:  fneer  36319  topfneec  36321  topfneec2  36322