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Theorem fneval 33760
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 5058 . . 3 (𝐴 𝐵𝐴(Fne ∩ Fne)𝐵)
3 brin 5104 . . . 4 (𝐴(Fne ∩ Fne)𝐵 ↔ (𝐴Fne𝐵𝐴Fne𝐵))
4 fnerel 33746 . . . . . 6 Rel Fne
54relbrcnv 5957 . . . . 5 (𝐴Fne𝐵𝐵Fne𝐴)
65anbi2i 625 . . . 4 ((𝐴Fne𝐵𝐴Fne𝐵) ↔ (𝐴Fne𝐵𝐵Fne𝐴))
73, 6bitri 278 . . 3 (𝐴(Fne ∩ Fne)𝐵 ↔ (𝐴Fne𝐵𝐵Fne𝐴))
82, 7bitri 278 . 2 (𝐴 𝐵 ↔ (𝐴Fne𝐵𝐵Fne𝐴))
9 eqid 2824 . . . . . 6 𝐴 = 𝐴
10 eqid 2824 . . . . . 6 𝐵 = 𝐵
119, 10isfne4b 33749 . . . . 5 (𝐵𝑊 → (𝐴Fne𝐵 ↔ ( 𝐴 = 𝐵 ∧ (topGen‘𝐴) ⊆ (topGen‘𝐵))))
1210, 9isfne4b 33749 . . . . . 6 (𝐴𝑉 → (𝐵Fne𝐴 ↔ ( 𝐵 = 𝐴 ∧ (topGen‘𝐵) ⊆ (topGen‘𝐴))))
13 eqcom 2831 . . . . . . 7 ( 𝐵 = 𝐴 𝐴 = 𝐵)
1413anbi1i 626 . . . . . 6 (( 𝐵 = 𝐴 ∧ (topGen‘𝐵) ⊆ (topGen‘𝐴)) ↔ ( 𝐴 = 𝐵 ∧ (topGen‘𝐵) ⊆ (topGen‘𝐴)))
1512, 14syl6bb 290 . . . . 5 (𝐴𝑉 → (𝐵Fne𝐴 ↔ ( 𝐴 = 𝐵 ∧ (topGen‘𝐵) ⊆ (topGen‘𝐴))))
1611, 15bi2anan9r 639 . . . 4 ((𝐴𝑉𝐵𝑊) → ((𝐴Fne𝐵𝐵Fne𝐴) ↔ (( 𝐴 = 𝐵 ∧ (topGen‘𝐴) ⊆ (topGen‘𝐵)) ∧ ( 𝐴 = 𝐵 ∧ (topGen‘𝐵) ⊆ (topGen‘𝐴)))))
17 eqss 3968 . . . . . 6 ((topGen‘𝐴) = (topGen‘𝐵) ↔ ((topGen‘𝐴) ⊆ (topGen‘𝐵) ∧ (topGen‘𝐵) ⊆ (topGen‘𝐴)))
1817anbi2i 625 . . . . 5 (( 𝐴 = 𝐵 ∧ (topGen‘𝐴) = (topGen‘𝐵)) ↔ ( 𝐴 = 𝐵 ∧ ((topGen‘𝐴) ⊆ (topGen‘𝐵) ∧ (topGen‘𝐵) ⊆ (topGen‘𝐴))))
19 anandi 675 . . . . 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 278 . . . 4 (( 𝐴 = 𝐵 ∧ (topGen‘𝐴) = (topGen‘𝐵)) ↔ (( 𝐴 = 𝐵 ∧ (topGen‘𝐴) ⊆ (topGen‘𝐵)) ∧ ( 𝐴 = 𝐵 ∧ (topGen‘𝐵) ⊆ (topGen‘𝐴))))
2116, 20syl6bbr 292 . . 3 ((𝐴𝑉𝐵𝑊) → ((𝐴Fne𝐵𝐵Fne𝐴) ↔ ( 𝐴 = 𝐵 ∧ (topGen‘𝐴) = (topGen‘𝐵))))
22 unieq 4835 . . . . 5 ((topGen‘𝐴) = (topGen‘𝐵) → (topGen‘𝐴) = (topGen‘𝐵))
23 unitg 21578 . . . . . 6 (𝐴𝑉 (topGen‘𝐴) = 𝐴)
24 unitg 21578 . . . . . 6 (𝐵𝑊 (topGen‘𝐵) = 𝐵)
2523, 24eqeqan12d 2841 . . . . 5 ((𝐴𝑉𝐵𝑊) → ( (topGen‘𝐴) = (topGen‘𝐵) ↔ 𝐴 = 𝐵))
2622, 25syl5ib 247 . . . 4 ((𝐴𝑉𝐵𝑊) → ((topGen‘𝐴) = (topGen‘𝐵) → 𝐴 = 𝐵))
2726pm4.71rd 566 . . 3 ((𝐴𝑉𝐵𝑊) → ((topGen‘𝐴) = (topGen‘𝐵) ↔ ( 𝐴 = 𝐵 ∧ (topGen‘𝐴) = (topGen‘𝐵))))
2821, 27bitr4d 285 . 2 ((𝐴𝑉𝐵𝑊) → ((𝐴Fne𝐵𝐵Fne𝐴) ↔ (topGen‘𝐴) = (topGen‘𝐵)))
298, 28syl5bb 286 1 ((𝐴𝑉𝐵𝑊) → (𝐴 𝐵 ↔ (topGen‘𝐴) = (topGen‘𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1538  wcel 2115  cin 3918  wss 3919   cuni 4824   class class class wbr 5052  ccnv 5541  cfv 6343  topGenctg 16711  Fnecfne 33744
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 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7455
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-sbc 3759  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-iun 4907  df-br 5053  df-opab 5115  df-mpt 5133  df-id 5447  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-iota 6302  df-fun 6345  df-fv 6351  df-topgen 16717  df-fne 33745
This theorem is referenced by:  fneer  33761  topfneec  33763  topfneec2  33764
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