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Theorem isfne4b 36343
Description: A condition for a topology to be finer than another. (Contributed by Jeff Hankins, 28-Sep-2009.) (Revised by Mario Carneiro, 11-Sep-2015.)
Hypotheses
Ref Expression
isfne.1 𝑋 = 𝐴
isfne.2 𝑌 = 𝐵
Assertion
Ref Expression
isfne4b (𝐵𝑉 → (𝐴Fne𝐵 ↔ (𝑋 = 𝑌 ∧ (topGen‘𝐴) ⊆ (topGen‘𝐵))))

Proof of Theorem isfne4b
StepHypRef Expression
1 isfne.1 . . 3 𝑋 = 𝐴
2 isfne.2 . . 3 𝑌 = 𝐵
31, 2isfne4 36342 . 2 (𝐴Fne𝐵 ↔ (𝑋 = 𝑌𝐴 ⊆ (topGen‘𝐵)))
4 simpr 484 . . . . . . 7 ((𝐵𝑉𝑋 = 𝑌) → 𝑋 = 𝑌)
54, 1, 23eqtr3g 2799 . . . . . 6 ((𝐵𝑉𝑋 = 𝑌) → 𝐴 = 𝐵)
6 uniexg 7761 . . . . . . 7 (𝐵𝑉 𝐵 ∈ V)
76adantr 480 . . . . . 6 ((𝐵𝑉𝑋 = 𝑌) → 𝐵 ∈ V)
85, 7eqeltrd 2840 . . . . 5 ((𝐵𝑉𝑋 = 𝑌) → 𝐴 ∈ V)
9 uniexb 7785 . . . . 5 (𝐴 ∈ V ↔ 𝐴 ∈ V)
108, 9sylibr 234 . . . 4 ((𝐵𝑉𝑋 = 𝑌) → 𝐴 ∈ V)
11 simpl 482 . . . 4 ((𝐵𝑉𝑋 = 𝑌) → 𝐵𝑉)
12 tgss3 22994 . . . 4 ((𝐴 ∈ V ∧ 𝐵𝑉) → ((topGen‘𝐴) ⊆ (topGen‘𝐵) ↔ 𝐴 ⊆ (topGen‘𝐵)))
1310, 11, 12syl2anc 584 . . 3 ((𝐵𝑉𝑋 = 𝑌) → ((topGen‘𝐴) ⊆ (topGen‘𝐵) ↔ 𝐴 ⊆ (topGen‘𝐵)))
1413pm5.32da 579 . 2 (𝐵𝑉 → ((𝑋 = 𝑌 ∧ (topGen‘𝐴) ⊆ (topGen‘𝐵)) ↔ (𝑋 = 𝑌𝐴 ⊆ (topGen‘𝐵))))
153, 14bitr4id 290 1 (𝐵𝑉 → (𝐴Fne𝐵 ↔ (𝑋 = 𝑌 ∧ (topGen‘𝐴) ⊆ (topGen‘𝐵))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1539  wcel 2107  Vcvv 3479  wss 3950   cuni 4906   class class class wbr 5142  cfv 6560  topGenctg 17483  Fnecfne 36338
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-iota 6513  df-fun 6562  df-fv 6568  df-topgen 17489  df-fne 36339
This theorem is referenced by:  fnetr  36353  fneval  36354
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