Theorem fneval 34537

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 5085	. . 3 ⊢ (𝐴 ∼ 𝐵 ↔ 𝐴(Fne ∩ ◡Fne)𝐵)
3		brin 5131	. . . 4 ⊢ (𝐴(Fne ∩ ◡Fne)𝐵 ↔ (𝐴Fne𝐵 ∧ 𝐴◡Fne𝐵))
4		fnerel 34523	. . . . . 6 ⊢ Rel Fne
5	4	relbrcnv 6014	. . . . 5 ⊢ (𝐴◡Fne𝐵 ↔ 𝐵Fne𝐴)
6	5	anbi2i 623	. . . 4 ⊢ ((𝐴Fne𝐵 ∧ 𝐴◡Fne𝐵) ↔ (𝐴Fne𝐵 ∧ 𝐵Fne𝐴))
7	3, 6	bitri 274	. . 3 ⊢ (𝐴(Fne ∩ ◡Fne)𝐵 ↔ (𝐴Fne𝐵 ∧ 𝐵Fne𝐴))
8	2, 7	bitri 274	. 2 ⊢ (𝐴 ∼ 𝐵 ↔ (𝐴Fne𝐵 ∧ 𝐵Fne𝐴))
9		eqid 2740	. . . . . 6 ⊢ ∪ 𝐴 = ∪ 𝐴
10		eqid 2740	. . . . . 6 ⊢ ∪ 𝐵 = ∪ 𝐵
11	9, 10	isfne4b 34526	. . . . 5 ⊢ (𝐵 ∈ 𝑊 → (𝐴Fne𝐵 ↔ (∪ 𝐴 = ∪ 𝐵 ∧ (topGen‘𝐴) ⊆ (topGen‘𝐵))))
12	10, 9	isfne4b 34526	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → (𝐵Fne𝐴 ↔ (∪ 𝐵 = ∪ 𝐴 ∧ (topGen‘𝐵) ⊆ (topGen‘𝐴))))
13		eqcom 2747	. . . . . . 7 ⊢ (∪ 𝐵 = ∪ 𝐴 ↔ ∪ 𝐴 = ∪ 𝐵)
14	13	anbi1i 624	. . . . . 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 637	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((𝐴Fne𝐵 ∧ 𝐵Fne𝐴) ↔ ((∪ 𝐴 = ∪ 𝐵 ∧ (topGen‘𝐴) ⊆ (topGen‘𝐵)) ∧ (∪ 𝐴 = ∪ 𝐵 ∧ (topGen‘𝐵) ⊆ (topGen‘𝐴)))))
17		eqss 3941	. . . . . 6 ⊢ ((topGen‘𝐴) = (topGen‘𝐵) ↔ ((topGen‘𝐴) ⊆ (topGen‘𝐵) ∧ (topGen‘𝐵) ⊆ (topGen‘𝐴)))
18	17	anbi2i 623	. . . . 5 ⊢ ((∪ 𝐴 = ∪ 𝐵 ∧ (topGen‘𝐴) = (topGen‘𝐵)) ↔ (∪ 𝐴 = ∪ 𝐵 ∧ ((topGen‘𝐴) ⊆ (topGen‘𝐵) ∧ (topGen‘𝐵) ⊆ (topGen‘𝐴))))
19		anandi 673	. . . . 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 274	. . . 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 4856	. . . . 5 ⊢ ((topGen‘𝐴) = (topGen‘𝐵) → ∪ (topGen‘𝐴) = ∪ (topGen‘𝐵))
23		unitg 22115	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → ∪ (topGen‘𝐴) = ∪ 𝐴)
24		unitg 22115	. . . . . 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	syl5ib 243	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((topGen‘𝐴) = (topGen‘𝐵) → ∪ 𝐴 = ∪ 𝐵))
27	26	pm4.71rd 563	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((topGen‘𝐴) = (topGen‘𝐵) ↔ (∪ 𝐴 = ∪ 𝐵 ∧ (topGen‘𝐴) = (topGen‘𝐵))))
28	21, 27	bitr4d 281	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((𝐴Fne𝐵 ∧ 𝐵Fne𝐴) ↔ (topGen‘𝐴) = (topGen‘𝐵)))
29	8, 28	syl5bb 283	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ∼ 𝐵 ↔ (topGen‘𝐴) = (topGen‘𝐵)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1542   ∈ wcel 2110   ∩ cin 3891   ⊆ wss 3892  ∪ cuni 4845   class class class wbr 5079  ^◡ccnv 5589  ‘cfv 6432  topGenctg 17146  Fnecfne 34521

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 1975  ax-7 2015  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2711  ax-sep 5227  ax-nul 5234  ax-pow 5292  ax-pr 5356  ax-un 7582

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2072  df-mo 2542  df-eu 2571  df-clab 2718  df-cleq 2732  df-clel 2818  df-nfc 2891  df-ral 3071  df-rex 3072  df-rab 3075  df-v 3433  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-nul 4263  df-if 4466  df-pw 4541  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4846  df-iun 4932  df-br 5080  df-opab 5142  df-mpt 5163  df-id 5490  df-xp 5596  df-rel 5597  df-cnv 5598  df-co 5599  df-dm 5600  df-iota 6390  df-fun 6434  df-fv 6440  df-topgen 17152  df-fne 34522

This theorem is referenced by:  fneer  34538  topfneec  34540  topfneec2  34541