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Theorem isfne4b 35214
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 = βˆͺ 𝐴
isfne.2 π‘Œ = βˆͺ 𝐡
Assertion
Ref Expression
isfne4b (𝐡 ∈ 𝑉 β†’ (𝐴Fne𝐡 ↔ (𝑋 = π‘Œ ∧ (topGenβ€˜π΄) βŠ† (topGenβ€˜π΅))))
Proof of Theorem isfne4b
StepHypRef Expression
1 isfne.1 . . 3 𝑋 = βˆͺ 𝐴
2 isfne.2 . . 3 π‘Œ = βˆͺ 𝐡
31, 2isfne4 35213 . 2 (𝐴Fne𝐡 ↔ (𝑋 = π‘Œ ∧ 𝐴 βŠ† (topGenβ€˜π΅)))
4 simpr 485 . . . . . . 7 ((𝐡 ∈ 𝑉 ∧ 𝑋 = π‘Œ) β†’ 𝑋 = π‘Œ)
54, 1, 23eqtr3g 2795 . . . . . 6 ((𝐡 ∈ 𝑉 ∧ 𝑋 = π‘Œ) β†’ βˆͺ 𝐴 = βˆͺ 𝐡)
6 uniexg 7726 . . . . . . 7 (𝐡 ∈ 𝑉 β†’ βˆͺ 𝐡 ∈ V)
76adantr 481 . . . . . 6 ((𝐡 ∈ 𝑉 ∧ 𝑋 = π‘Œ) β†’ βˆͺ 𝐡 ∈ V)
85, 7eqeltrd 2833 . . . . 5 ((𝐡 ∈ 𝑉 ∧ 𝑋 = π‘Œ) β†’ βˆͺ 𝐴 ∈ V)
9 uniexb 7747 . . . . 5 (𝐴 ∈ V ↔ βˆͺ 𝐴 ∈ V)
108, 9sylibr 233 . . . 4 ((𝐡 ∈ 𝑉 ∧ 𝑋 = π‘Œ) β†’ 𝐴 ∈ V)
11 simpl 483 . . . 4 ((𝐡 ∈ 𝑉 ∧ 𝑋 = π‘Œ) β†’ 𝐡 ∈ 𝑉)
12 tgss3 22480 . . . 4 ((𝐴 ∈ V ∧ 𝐡 ∈ 𝑉) β†’ ((topGenβ€˜π΄) βŠ† (topGenβ€˜π΅) ↔ 𝐴 βŠ† (topGenβ€˜π΅)))
1310, 11, 12syl2anc 584 . . 3 ((𝐡 ∈ 𝑉 ∧ 𝑋 = π‘Œ) β†’ ((topGenβ€˜π΄) βŠ† (topGenβ€˜π΅) ↔ 𝐴 βŠ† (topGenβ€˜π΅)))
1413pm5.32da 579 . 2 (𝐡 ∈ 𝑉 β†’ ((𝑋 = π‘Œ ∧ (topGenβ€˜π΄) βŠ† (topGenβ€˜π΅)) ↔ (𝑋 = π‘Œ ∧ 𝐴 βŠ† (topGenβ€˜π΅))))
153, 14bitr4id 289 1 (𝐡 ∈ 𝑉 β†’ (𝐴Fne𝐡 ↔ (𝑋 = π‘Œ ∧ (topGenβ€˜π΄) βŠ† (topGenβ€˜π΅))))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 396   = wceq 1541   ∈ wcel 2106  Vcvv 3474   βŠ† wss 3947  βˆͺ cuni 4907   class class class wbr 5147  β€˜cfv 6540  topGenctg 17379  Fnecfne 35209
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-iota 6492  df-fun 6542  df-fv 6548  df-topgen 17385  df-fne 35210
This theorem is referenced by:  fnetr  35224  fneval  35225
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