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Theorem tgtop 22476
Description: A topology is its own basis. (Contributed by NM, 18-Jul-2006.)
Assertion
Ref Expression
tgtop (𝐽 ∈ Top β†’ (topGenβ€˜π½) = 𝐽)
Proof of Theorem tgtop
Dummy variables π‘₯ 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eltg3 22465 . . . 4 (𝐽 ∈ Top β†’ (π‘₯ ∈ (topGenβ€˜π½) ↔ βˆƒπ‘¦(𝑦 βŠ† 𝐽 ∧ π‘₯ = βˆͺ 𝑦)))
2 simpr 486 . . . . . . 7 (((𝐽 ∈ Top ∧ 𝑦 βŠ† 𝐽) ∧ π‘₯ = βˆͺ 𝑦) β†’ π‘₯ = βˆͺ 𝑦)
3 uniopn 22399 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝑦 βŠ† 𝐽) β†’ βˆͺ 𝑦 ∈ 𝐽)
43adantr 482 . . . . . . 7 (((𝐽 ∈ Top ∧ 𝑦 βŠ† 𝐽) ∧ π‘₯ = βˆͺ 𝑦) β†’ βˆͺ 𝑦 ∈ 𝐽)
52, 4eqeltrd 2834 . . . . . 6 (((𝐽 ∈ Top ∧ 𝑦 βŠ† 𝐽) ∧ π‘₯ = βˆͺ 𝑦) β†’ π‘₯ ∈ 𝐽)
65expl 459 . . . . 5 (𝐽 ∈ Top β†’ ((𝑦 βŠ† 𝐽 ∧ π‘₯ = βˆͺ 𝑦) β†’ π‘₯ ∈ 𝐽))
76exlimdv 1937 . . . 4 (𝐽 ∈ Top β†’ (βˆƒπ‘¦(𝑦 βŠ† 𝐽 ∧ π‘₯ = βˆͺ 𝑦) β†’ π‘₯ ∈ 𝐽))
81, 7sylbid 239 . . 3 (𝐽 ∈ Top β†’ (π‘₯ ∈ (topGenβ€˜π½) β†’ π‘₯ ∈ 𝐽))
98ssrdv 3989 . 2 (𝐽 ∈ Top β†’ (topGenβ€˜π½) βŠ† 𝐽)
10 bastg 22469 . 2 (𝐽 ∈ Top β†’ 𝐽 βŠ† (topGenβ€˜π½))
119, 10eqssd 4000 1 (𝐽 ∈ Top β†’ (topGenβ€˜π½) = 𝐽)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 397   = wceq 1542  βˆƒwex 1782   ∈ wcel 2107   βŠ† wss 3949  βˆͺ cuni 4909  β€˜cfv 6544  topGenctg 17383  Topctop 22395
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-iota 6496  df-fun 6546  df-fv 6552  df-topgen 17389  df-top 22396
This theorem is referenced by:  eltop  22477  eltop2  22478  eltop3  22479  bastop  22484  tgtop11  22485  basgen  22491  tgfiss  22494  bastop1  22496  resttop  22664  dis1stc  23003  alexsubALTlem1  23551  xrtgioo  24322  topfne  35239  topfneec  35240  topfneec2  35241  dissneqlem  36221
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