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Theorem fneval 35967
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 ((𝐴𝑉𝐵𝑊) → (𝐴 𝐵 ↔ (topGen‘𝐴) = (topGen‘𝐵)))

Proof of Theorem fneval
StepHypRef Expression
1 fneval.1 . . . 4 = (Fne ∩ Fne)
21breqi 5155 . . 3 (𝐴 𝐵𝐴(Fne ∩ Fne)𝐵)
3 brin 5201 . . . 4 (𝐴(Fne ∩ Fne)𝐵 ↔ (𝐴Fne𝐵𝐴Fne𝐵))
4 fnerel 35953 . . . . . 6 Rel Fne
54relbrcnv 6112 . . . . 5 (𝐴Fne𝐵𝐵Fne𝐴)
65anbi2i 621 . . . 4 ((𝐴Fne𝐵𝐴Fne𝐵) ↔ (𝐴Fne𝐵𝐵Fne𝐴))
73, 6bitri 274 . . 3 (𝐴(Fne ∩ Fne)𝐵 ↔ (𝐴Fne𝐵𝐵Fne𝐴))
82, 7bitri 274 . 2 (𝐴 𝐵 ↔ (𝐴Fne𝐵𝐵Fne𝐴))
9 eqid 2725 . . . . . 6 𝐴 = 𝐴
10 eqid 2725 . . . . . 6 𝐵 = 𝐵
119, 10isfne4b 35956 . . . . 5 (𝐵𝑊 → (𝐴Fne𝐵 ↔ ( 𝐴 = 𝐵 ∧ (topGen‘𝐴) ⊆ (topGen‘𝐵))))
1210, 9isfne4b 35956 . . . . . 6 (𝐴𝑉 → (𝐵Fne𝐴 ↔ ( 𝐵 = 𝐴 ∧ (topGen‘𝐵) ⊆ (topGen‘𝐴))))
13 eqcom 2732 . . . . . . 7 ( 𝐵 = 𝐴 𝐴 = 𝐵)
1413anbi1i 622 . . . . . 6 (( 𝐵 = 𝐴 ∧ (topGen‘𝐵) ⊆ (topGen‘𝐴)) ↔ ( 𝐴 = 𝐵 ∧ (topGen‘𝐵) ⊆ (topGen‘𝐴)))
1512, 14bitrdi 286 . . . . 5 (𝐴𝑉 → (𝐵Fne𝐴 ↔ ( 𝐴 = 𝐵 ∧ (topGen‘𝐵) ⊆ (topGen‘𝐴))))
1611, 15bi2anan9r 637 . . . 4 ((𝐴𝑉𝐵𝑊) → ((𝐴Fne𝐵𝐵Fne𝐴) ↔ (( 𝐴 = 𝐵 ∧ (topGen‘𝐴) ⊆ (topGen‘𝐵)) ∧ ( 𝐴 = 𝐵 ∧ (topGen‘𝐵) ⊆ (topGen‘𝐴)))))
17 eqss 3992 . . . . . 6 ((topGen‘𝐴) = (topGen‘𝐵) ↔ ((topGen‘𝐴) ⊆ (topGen‘𝐵) ∧ (topGen‘𝐵) ⊆ (topGen‘𝐴)))
1817anbi2i 621 . . . . 5 (( 𝐴 = 𝐵 ∧ (topGen‘𝐴) = (topGen‘𝐵)) ↔ ( 𝐴 = 𝐵 ∧ ((topGen‘𝐴) ⊆ (topGen‘𝐵) ∧ (topGen‘𝐵) ⊆ (topGen‘𝐴))))
19 anandi 674 . . . . 5 (( 𝐴 = 𝐵 ∧ ((topGen‘𝐴) ⊆ (topGen‘𝐵) ∧ (topGen‘𝐵) ⊆ (topGen‘𝐴))) ↔ (( 𝐴 = 𝐵 ∧ (topGen‘𝐴) ⊆ (topGen‘𝐵)) ∧ ( 𝐴 = 𝐵 ∧ (topGen‘𝐵) ⊆ (topGen‘𝐴))))
2018, 19bitri 274 . . . 4 (( 𝐴 = 𝐵 ∧ (topGen‘𝐴) = (topGen‘𝐵)) ↔ (( 𝐴 = 𝐵 ∧ (topGen‘𝐴) ⊆ (topGen‘𝐵)) ∧ ( 𝐴 = 𝐵 ∧ (topGen‘𝐵) ⊆ (topGen‘𝐴))))
2116, 20bitr4di 288 . . 3 ((𝐴𝑉𝐵𝑊) → ((𝐴Fne𝐵𝐵Fne𝐴) ↔ ( 𝐴 = 𝐵 ∧ (topGen‘𝐴) = (topGen‘𝐵))))
22 unieq 4920 . . . . 5 ((topGen‘𝐴) = (topGen‘𝐵) → (topGen‘𝐴) = (topGen‘𝐵))
23 unitg 22914 . . . . . 6 (𝐴𝑉 (topGen‘𝐴) = 𝐴)
24 unitg 22914 . . . . . 6 (𝐵𝑊 (topGen‘𝐵) = 𝐵)
2523, 24eqeqan12d 2739 . . . . 5 ((𝐴𝑉𝐵𝑊) → ( (topGen‘𝐴) = (topGen‘𝐵) ↔ 𝐴 = 𝐵))
2622, 25imbitrid 243 . . . 4 ((𝐴𝑉𝐵𝑊) → ((topGen‘𝐴) = (topGen‘𝐵) → 𝐴 = 𝐵))
2726pm4.71rd 561 . . 3 ((𝐴𝑉𝐵𝑊) → ((topGen‘𝐴) = (topGen‘𝐵) ↔ ( 𝐴 = 𝐵 ∧ (topGen‘𝐴) = (topGen‘𝐵))))
2821, 27bitr4d 281 . 2 ((𝐴𝑉𝐵𝑊) → ((𝐴Fne𝐵𝐵Fne𝐴) ↔ (topGen‘𝐴) = (topGen‘𝐵)))
298, 28bitrid 282 1 ((𝐴𝑉𝐵𝑊) → (𝐴 𝐵 ↔ (topGen‘𝐴) = (topGen‘𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394   = wceq 1533  wcel 2098  cin 3943  wss 3944   cuni 4909   class class class wbr 5149  ccnv 5677  cfv 6549  topGenctg 17422  Fnecfne 35951
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-un 7741
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-rab 3419  df-v 3463  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-iun 4999  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-iota 6501  df-fun 6551  df-fv 6557  df-topgen 17428  df-fne 35952
This theorem is referenced by:  fneer  35968  topfneec  35970  topfneec2  35971
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