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Theorem fneval 31986
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 4619 . . 3 (𝐴 𝐵𝐴(Fne ∩ Fne)𝐵)
3 brin 4664 . . . 4 (𝐴(Fne ∩ Fne)𝐵 ↔ (𝐴Fne𝐵𝐴Fne𝐵))
4 fnerel 31972 . . . . . 6 Rel Fne
54relbrcnv 5465 . . . . 5 (𝐴Fne𝐵𝐵Fne𝐴)
65anbi2i 729 . . . 4 ((𝐴Fne𝐵𝐴Fne𝐵) ↔ (𝐴Fne𝐵𝐵Fne𝐴))
73, 6bitri 264 . . 3 (𝐴(Fne ∩ Fne)𝐵 ↔ (𝐴Fne𝐵𝐵Fne𝐴))
82, 7bitri 264 . 2 (𝐴 𝐵 ↔ (𝐴Fne𝐵𝐵Fne𝐴))
9 eqid 2621 . . . . . 6 𝐴 = 𝐴
10 eqid 2621 . . . . . 6 𝐵 = 𝐵
119, 10isfne4b 31975 . . . . 5 (𝐵𝑊 → (𝐴Fne𝐵 ↔ ( 𝐴 = 𝐵 ∧ (topGen‘𝐴) ⊆ (topGen‘𝐵))))
1210, 9isfne4b 31975 . . . . . 6 (𝐴𝑉 → (𝐵Fne𝐴 ↔ ( 𝐵 = 𝐴 ∧ (topGen‘𝐵) ⊆ (topGen‘𝐴))))
13 eqcom 2628 . . . . . . 7 ( 𝐵 = 𝐴 𝐴 = 𝐵)
1413anbi1i 730 . . . . . 6 (( 𝐵 = 𝐴 ∧ (topGen‘𝐵) ⊆ (topGen‘𝐴)) ↔ ( 𝐴 = 𝐵 ∧ (topGen‘𝐵) ⊆ (topGen‘𝐴)))
1512, 14syl6bb 276 . . . . 5 (𝐴𝑉 → (𝐵Fne𝐴 ↔ ( 𝐴 = 𝐵 ∧ (topGen‘𝐵) ⊆ (topGen‘𝐴))))
1611, 15bi2anan9r 917 . . . 4 ((𝐴𝑉𝐵𝑊) → ((𝐴Fne𝐵𝐵Fne𝐴) ↔ (( 𝐴 = 𝐵 ∧ (topGen‘𝐴) ⊆ (topGen‘𝐵)) ∧ ( 𝐴 = 𝐵 ∧ (topGen‘𝐵) ⊆ (topGen‘𝐴)))))
17 eqss 3598 . . . . . 6 ((topGen‘𝐴) = (topGen‘𝐵) ↔ ((topGen‘𝐴) ⊆ (topGen‘𝐵) ∧ (topGen‘𝐵) ⊆ (topGen‘𝐴)))
1817anbi2i 729 . . . . 5 (( 𝐴 = 𝐵 ∧ (topGen‘𝐴) = (topGen‘𝐵)) ↔ ( 𝐴 = 𝐵 ∧ ((topGen‘𝐴) ⊆ (topGen‘𝐵) ∧ (topGen‘𝐵) ⊆ (topGen‘𝐴))))
19 anandi 870 . . . . 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 264 . . . 4 (( 𝐴 = 𝐵 ∧ (topGen‘𝐴) = (topGen‘𝐵)) ↔ (( 𝐴 = 𝐵 ∧ (topGen‘𝐴) ⊆ (topGen‘𝐵)) ∧ ( 𝐴 = 𝐵 ∧ (topGen‘𝐵) ⊆ (topGen‘𝐴))))
2116, 20syl6bbr 278 . . 3 ((𝐴𝑉𝐵𝑊) → ((𝐴Fne𝐵𝐵Fne𝐴) ↔ ( 𝐴 = 𝐵 ∧ (topGen‘𝐴) = (topGen‘𝐵))))
22 unieq 4410 . . . . 5 ((topGen‘𝐴) = (topGen‘𝐵) → (topGen‘𝐴) = (topGen‘𝐵))
23 unitg 20682 . . . . . 6 (𝐴𝑉 (topGen‘𝐴) = 𝐴)
24 unitg 20682 . . . . . 6 (𝐵𝑊 (topGen‘𝐵) = 𝐵)
2523, 24eqeqan12d 2637 . . . . 5 ((𝐴𝑉𝐵𝑊) → ( (topGen‘𝐴) = (topGen‘𝐵) ↔ 𝐴 = 𝐵))
2622, 25syl5ib 234 . . . 4 ((𝐴𝑉𝐵𝑊) → ((topGen‘𝐴) = (topGen‘𝐵) → 𝐴 = 𝐵))
2726pm4.71rd 666 . . 3 ((𝐴𝑉𝐵𝑊) → ((topGen‘𝐴) = (topGen‘𝐵) ↔ ( 𝐴 = 𝐵 ∧ (topGen‘𝐴) = (topGen‘𝐵))))
2821, 27bitr4d 271 . 2 ((𝐴𝑉𝐵𝑊) → ((𝐴Fne𝐵𝐵Fne𝐴) ↔ (topGen‘𝐴) = (topGen‘𝐵)))
298, 28syl5bb 272 1 ((𝐴𝑉𝐵𝑊) → (𝐴 𝐵 ↔ (topGen‘𝐴) = (topGen‘𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1480  wcel 1987  cin 3554  wss 3555   cuni 4402   class class class wbr 4613  ccnv 5073  cfv 5847  topGenctg 16019  Fnecfne 31970
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-iota 5810  df-fun 5849  df-fv 5855  df-topgen 16025  df-fne 31971
This theorem is referenced by:  fneer  31987  topfneec  31989  topfneec2  31990
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