Theorem fneval 36580

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 5078	. . 3 ⊢ (𝐴 ∼ 𝐵 ↔ 𝐴(Fne ∩ ◡Fne)𝐵)
3		brin 5124	. . . 4 ⊢ (𝐴(Fne ∩ ◡Fne)𝐵 ↔ (𝐴Fne𝐵 ∧ 𝐴◡Fne𝐵))
4		fnerel 36566	. . . . . 6 ⊢ Rel Fne
5	4	relbrcnv 6059	. . . . 5 ⊢ (𝐴◡Fne𝐵 ↔ 𝐵Fne𝐴)
6	5	anbi2i 629	. . . 4 ⊢ ((𝐴Fne𝐵 ∧ 𝐴◡Fne𝐵) ↔ (𝐴Fne𝐵 ∧ 𝐵Fne𝐴))
7	3, 6	bitri 276	. . 3 ⊢ (𝐴(Fne ∩ ◡Fne)𝐵 ↔ (𝐴Fne𝐵 ∧ 𝐵Fne𝐴))
8	2, 7	bitri 276	. 2 ⊢ (𝐴 ∼ 𝐵 ↔ (𝐴Fne𝐵 ∧ 𝐵Fne𝐴))
9		eqid 2739	. . . . . 6 ⊢ ∪ 𝐴 = ∪ 𝐴
10		eqid 2739	. . . . . 6 ⊢ ∪ 𝐵 = ∪ 𝐵
11	9, 10	isfne4b 36569	. . . . 5 ⊢ (𝐵 ∈ 𝑊 → (𝐴Fne𝐵 ↔ (∪ 𝐴 = ∪ 𝐵 ∧ (topGen‘𝐴) ⊆ (topGen‘𝐵))))
12	10, 9	isfne4b 36569	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → (𝐵Fne𝐴 ↔ (∪ 𝐵 = ∪ 𝐴 ∧ (topGen‘𝐵) ⊆ (topGen‘𝐴))))
13		eqcom 2746	. . . . . . 7 ⊢ (∪ 𝐵 = ∪ 𝐴 ↔ ∪ 𝐴 = ∪ 𝐵)
14	13	anbi1i 630	. . . . . 6 ⊢ ((∪ 𝐵 = ∪ 𝐴 ∧ (topGen‘𝐵) ⊆ (topGen‘𝐴)) ↔ (∪ 𝐴 = ∪ 𝐵 ∧ (topGen‘𝐵) ⊆ (topGen‘𝐴)))
15	12, 14	bitrdi 288	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → (𝐵Fne𝐴 ↔ (∪ 𝐴 = ∪ 𝐵 ∧ (topGen‘𝐵) ⊆ (topGen‘𝐴))))
16	11, 15	bi2anan9r 645	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((𝐴Fne𝐵 ∧ 𝐵Fne𝐴) ↔ ((∪ 𝐴 = ∪ 𝐵 ∧ (topGen‘𝐴) ⊆ (topGen‘𝐵)) ∧ (∪ 𝐴 = ∪ 𝐵 ∧ (topGen‘𝐵) ⊆ (topGen‘𝐴)))))
17		eqss 3930	. . . . . 6 ⊢ ((topGen‘𝐴) = (topGen‘𝐵) ↔ ((topGen‘𝐴) ⊆ (topGen‘𝐵) ∧ (topGen‘𝐵) ⊆ (topGen‘𝐴)))
18	17	anbi2i 629	. . . . 5 ⊢ ((∪ 𝐴 = ∪ 𝐵 ∧ (topGen‘𝐴) = (topGen‘𝐵)) ↔ (∪ 𝐴 = ∪ 𝐵 ∧ ((topGen‘𝐴) ⊆ (topGen‘𝐵) ∧ (topGen‘𝐵) ⊆ (topGen‘𝐴))))
19		anandi 682	. . . . 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 276	. . . 4 ⊢ ((∪ 𝐴 = ∪ 𝐵 ∧ (topGen‘𝐴) = (topGen‘𝐵)) ↔ ((∪ 𝐴 = ∪ 𝐵 ∧ (topGen‘𝐴) ⊆ (topGen‘𝐵)) ∧ (∪ 𝐴 = ∪ 𝐵 ∧ (topGen‘𝐵) ⊆ (topGen‘𝐴))))
21	16, 20	bitr4di 290	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((𝐴Fne𝐵 ∧ 𝐵Fne𝐴) ↔ (∪ 𝐴 = ∪ 𝐵 ∧ (topGen‘𝐴) = (topGen‘𝐵))))
22		unieq 4849	. . . . 5 ⊢ ((topGen‘𝐴) = (topGen‘𝐵) → ∪ (topGen‘𝐴) = ∪ (topGen‘𝐵))
23		unitg 22950	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → ∪ (topGen‘𝐴) = ∪ 𝐴)
24		unitg 22950	. . . . . 6 ⊢ (𝐵 ∈ 𝑊 → ∪ (topGen‘𝐵) = ∪ 𝐵)
25	23, 24	eqeqan12d 2753	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (∪ (topGen‘𝐴) = ∪ (topGen‘𝐵) ↔ ∪ 𝐴 = ∪ 𝐵))
26	22, 25	imbitrid 245	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((topGen‘𝐴) = (topGen‘𝐵) → ∪ 𝐴 = ∪ 𝐵))
27	26	pm4.71rd 567	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((topGen‘𝐴) = (topGen‘𝐵) ↔ (∪ 𝐴 = ∪ 𝐵 ∧ (topGen‘𝐴) = (topGen‘𝐵))))
28	21, 27	bitr4d 283	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((𝐴Fne𝐵 ∧ 𝐵Fne𝐴) ↔ (topGen‘𝐴) = (topGen‘𝐵)))
29	8, 28	bitrid 284	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ∼ 𝐵 ↔ (topGen‘𝐴) = (topGen‘𝐵)))

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   = wceq 1547   ∈ wcel 2119   ∩ cin 3882   ⊆ wss 3883  ∪ cuni 4838   class class class wbr 5072  ^◡ccnv 5617  ‘cfv 6485  topGenctg 17391  Fnecfne 36564

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-sep 5218  ax-nul 5228  ax-pow 5294  ax-pr 5362  ax-un 7678

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-ral 3054  df-rex 3064  df-rab 3392  df-v 3433  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-iun 4923  df-br 5073  df-opab 5135  df-mpt 5154  df-id 5513  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-iota 6441  df-fun 6487  df-fv 6493  df-topgen 17397  df-fne 36565

This theorem is referenced by:  fneer  36581  topfneec  36583  topfneec2  36584