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Theorem bastop2 22915
Description: A version of bastop1 22914 that doesn't have 𝐡 βŠ† 𝐽 in the antecedent. (Contributed by NM, 3-Feb-2008.)
Assertion
Ref Expression
bastop2 (𝐽 ∈ Top β†’ ((topGenβ€˜π΅) = 𝐽 ↔ (𝐡 βŠ† 𝐽 ∧ βˆ€π‘₯ ∈ 𝐽 βˆƒπ‘¦(𝑦 βŠ† 𝐡 ∧ π‘₯ = βˆͺ 𝑦))))
Distinct variable groups:   π‘₯,𝑦,𝐡   π‘₯,𝐽,𝑦
Proof of Theorem bastop2
StepHypRef Expression
1 eleq1 2816 . . . . . . . 8 ((topGenβ€˜π΅) = 𝐽 β†’ ((topGenβ€˜π΅) ∈ Top ↔ 𝐽 ∈ Top))
21biimparc 478 . . . . . . 7 ((𝐽 ∈ Top ∧ (topGenβ€˜π΅) = 𝐽) β†’ (topGenβ€˜π΅) ∈ Top)
3 tgclb 22891 . . . . . . 7 (𝐡 ∈ TopBases ↔ (topGenβ€˜π΅) ∈ Top)
42, 3sylibr 233 . . . . . 6 ((𝐽 ∈ Top ∧ (topGenβ€˜π΅) = 𝐽) β†’ 𝐡 ∈ TopBases)
5 bastg 22887 . . . . . 6 (𝐡 ∈ TopBases β†’ 𝐡 βŠ† (topGenβ€˜π΅))
64, 5syl 17 . . . . 5 ((𝐽 ∈ Top ∧ (topGenβ€˜π΅) = 𝐽) β†’ 𝐡 βŠ† (topGenβ€˜π΅))
7 simpr 483 . . . . 5 ((𝐽 ∈ Top ∧ (topGenβ€˜π΅) = 𝐽) β†’ (topGenβ€˜π΅) = 𝐽)
86, 7sseqtrd 4020 . . . 4 ((𝐽 ∈ Top ∧ (topGenβ€˜π΅) = 𝐽) β†’ 𝐡 βŠ† 𝐽)
98ex 411 . . 3 (𝐽 ∈ Top β†’ ((topGenβ€˜π΅) = 𝐽 β†’ 𝐡 βŠ† 𝐽))
109pm4.71rd 561 . 2 (𝐽 ∈ Top β†’ ((topGenβ€˜π΅) = 𝐽 ↔ (𝐡 βŠ† 𝐽 ∧ (topGenβ€˜π΅) = 𝐽)))
11 bastop1 22914 . . 3 ((𝐽 ∈ Top ∧ 𝐡 βŠ† 𝐽) β†’ ((topGenβ€˜π΅) = 𝐽 ↔ βˆ€π‘₯ ∈ 𝐽 βˆƒπ‘¦(𝑦 βŠ† 𝐡 ∧ π‘₯ = βˆͺ 𝑦)))
1211pm5.32da 577 . 2 (𝐽 ∈ Top β†’ ((𝐡 βŠ† 𝐽 ∧ (topGenβ€˜π΅) = 𝐽) ↔ (𝐡 βŠ† 𝐽 ∧ βˆ€π‘₯ ∈ 𝐽 βˆƒπ‘¦(𝑦 βŠ† 𝐡 ∧ π‘₯ = βˆͺ 𝑦))))
1310, 12bitrd 278 1 (𝐽 ∈ Top β†’ ((topGenβ€˜π΅) = 𝐽 ↔ (𝐡 βŠ† 𝐽 ∧ βˆ€π‘₯ ∈ 𝐽 βˆƒπ‘¦(𝑦 βŠ† 𝐡 ∧ π‘₯ = βˆͺ 𝑦))))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 394   = wceq 1533  βˆƒwex 1773   ∈ wcel 2098  βˆ€wral 3057   βŠ† wss 3947  βˆͺ cuni 4910  β€˜cfv 6551  topGenctg 17424  Topctop 22813  TopBasesctb 22866
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 2698  ax-sep 5301  ax-nul 5308  ax-pow 5367  ax-pr 5431  ax-un 7744
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 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2937  df-ral 3058  df-rex 3067  df-rab 3429  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4325  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4911  df-br 5151  df-opab 5213  df-mpt 5234  df-id 5578  df-xp 5686  df-rel 5687  df-cnv 5688  df-co 5689  df-dm 5690  df-iota 6503  df-fun 6553  df-fv 6559  df-topgen 17430  df-top 22814  df-bases 22867
This theorem is referenced by: (None)
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