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Theorem isfne4b 35858
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 35857 . 2 (𝐴Fne𝐡 ↔ (𝑋 = π‘Œ ∧ 𝐴 βŠ† (topGenβ€˜π΅)))
4 simpr 483 . . . . . . 7 ((𝐡 ∈ 𝑉 ∧ 𝑋 = π‘Œ) β†’ 𝑋 = π‘Œ)
54, 1, 23eqtr3g 2791 . . . . . 6 ((𝐡 ∈ 𝑉 ∧ 𝑋 = π‘Œ) β†’ βˆͺ 𝐴 = βˆͺ 𝐡)
6 uniexg 7751 . . . . . . 7 (𝐡 ∈ 𝑉 β†’ βˆͺ 𝐡 ∈ V)
76adantr 479 . . . . . 6 ((𝐡 ∈ 𝑉 ∧ 𝑋 = π‘Œ) β†’ βˆͺ 𝐡 ∈ V)
85, 7eqeltrd 2829 . . . . 5 ((𝐡 ∈ 𝑉 ∧ 𝑋 = π‘Œ) β†’ βˆͺ 𝐴 ∈ V)
9 uniexb 7772 . . . . 5 (𝐴 ∈ V ↔ βˆͺ 𝐴 ∈ V)
108, 9sylibr 233 . . . 4 ((𝐡 ∈ 𝑉 ∧ 𝑋 = π‘Œ) β†’ 𝐴 ∈ V)
11 simpl 481 . . . 4 ((𝐡 ∈ 𝑉 ∧ 𝑋 = π‘Œ) β†’ 𝐡 ∈ 𝑉)
12 tgss3 22909 . . . 4 ((𝐴 ∈ V ∧ 𝐡 ∈ 𝑉) β†’ ((topGenβ€˜π΄) βŠ† (topGenβ€˜π΅) ↔ 𝐴 βŠ† (topGenβ€˜π΅)))
1310, 11, 12syl2anc 582 . . 3 ((𝐡 ∈ 𝑉 ∧ 𝑋 = π‘Œ) β†’ ((topGenβ€˜π΄) βŠ† (topGenβ€˜π΅) ↔ 𝐴 βŠ† (topGenβ€˜π΅)))
1413pm5.32da 577 . 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 394   = wceq 1533   ∈ wcel 2098  Vcvv 3473   βŠ† wss 3949  βˆͺ cuni 4912   class class class wbr 5152  β€˜cfv 6553  topGenctg 17426  Fnecfne 35853
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 2699  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7746
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 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-iota 6505  df-fun 6555  df-fv 6561  df-topgen 17432  df-fne 35854
This theorem is referenced by:  fnetr  35868  fneval  35869
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