Theorem fneval 36568

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 5106	. . 3 ⊢ (𝐴 ∼ 𝐵 ↔ 𝐴(Fne ∩ ◡Fne)𝐵)
3		brin 5152	. . . 4 ⊢ (𝐴(Fne ∩ ◡Fne)𝐵 ↔ (𝐴Fne𝐵 ∧ 𝐴◡Fne𝐵))
4		fnerel 36554	. . . . . 6 ⊢ Rel Fne
5	4	relbrcnv 6074	. . . . 5 ⊢ (𝐴◡Fne𝐵 ↔ 𝐵Fne𝐴)
6	5	anbi2i 624	. . . 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 2737	. . . . . 6 ⊢ ∪ 𝐴 = ∪ 𝐴
10		eqid 2737	. . . . . 6 ⊢ ∪ 𝐵 = ∪ 𝐵
11	9, 10	isfne4b 36557	. . . . 5 ⊢ (𝐵 ∈ 𝑊 → (𝐴Fne𝐵 ↔ (∪ 𝐴 = ∪ 𝐵 ∧ (topGen‘𝐴) ⊆ (topGen‘𝐵))))
12	10, 9	isfne4b 36557	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → (𝐵Fne𝐴 ↔ (∪ 𝐵 = ∪ 𝐴 ∧ (topGen‘𝐵) ⊆ (topGen‘𝐴))))
13		eqcom 2744	. . . . . . 7 ⊢ (∪ 𝐵 = ∪ 𝐴 ↔ ∪ 𝐴 = ∪ 𝐵)
14	13	anbi1i 625	. . . . . 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 640	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((𝐴Fne𝐵 ∧ 𝐵Fne𝐴) ↔ ((∪ 𝐴 = ∪ 𝐵 ∧ (topGen‘𝐴) ⊆ (topGen‘𝐵)) ∧ (∪ 𝐴 = ∪ 𝐵 ∧ (topGen‘𝐵) ⊆ (topGen‘𝐴)))))
17		eqss 3951	. . . . . 6 ⊢ ((topGen‘𝐴) = (topGen‘𝐵) ↔ ((topGen‘𝐴) ⊆ (topGen‘𝐵) ∧ (topGen‘𝐵) ⊆ (topGen‘𝐴)))
18	17	anbi2i 624	. . . . 5 ⊢ ((∪ 𝐴 = ∪ 𝐵 ∧ (topGen‘𝐴) = (topGen‘𝐵)) ↔ (∪ 𝐴 = ∪ 𝐵 ∧ ((topGen‘𝐴) ⊆ (topGen‘𝐵) ∧ (topGen‘𝐵) ⊆ (topGen‘𝐴))))
19		anandi 677	. . . . 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 4876	. . . . 5 ⊢ ((topGen‘𝐴) = (topGen‘𝐵) → ∪ (topGen‘𝐴) = ∪ (topGen‘𝐵))
23		unitg 22923	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → ∪ (topGen‘𝐴) = ∪ 𝐴)
24		unitg 22923	. . . . . 6 ⊢ (𝐵 ∈ 𝑊 → ∪ (topGen‘𝐵) = ∪ 𝐵)
25	23, 24	eqeqan12d 2751	. . . . 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 1542   ∈ wcel 2114   ∩ cin 3902   ⊆ wss 3903  ∪ cuni 4865   class class class wbr 5100  ^◡ccnv 5631  ‘cfv 6500  topGenctg 17369  Fnecfne 36552

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-iota 6456  df-fun 6502  df-fv 6508  df-topgen 17375  df-fne 36553

This theorem is referenced by:  fneer  36569  topfneec  36571  topfneec2  36572