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Theorem tgtop 22981
Description: A topology is its own basis. (Contributed by NM, 18-Jul-2006.)
Assertion
Ref Expression
tgtop (𝐽 ∈ Top → (topGen‘𝐽) = 𝐽)

Proof of Theorem tgtop
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eltg3 22970 . . . 4 (𝐽 ∈ Top → (𝑥 ∈ (topGen‘𝐽) ↔ ∃𝑦(𝑦𝐽𝑥 = 𝑦)))
2 simpr 484 . . . . . . 7 (((𝐽 ∈ Top ∧ 𝑦𝐽) ∧ 𝑥 = 𝑦) → 𝑥 = 𝑦)
3 uniopn 22904 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝑦𝐽) → 𝑦𝐽)
43adantr 480 . . . . . . 7 (((𝐽 ∈ Top ∧ 𝑦𝐽) ∧ 𝑥 = 𝑦) → 𝑦𝐽)
52, 4eqeltrd 2840 . . . . . 6 (((𝐽 ∈ Top ∧ 𝑦𝐽) ∧ 𝑥 = 𝑦) → 𝑥𝐽)
65expl 457 . . . . 5 (𝐽 ∈ Top → ((𝑦𝐽𝑥 = 𝑦) → 𝑥𝐽))
76exlimdv 1932 . . . 4 (𝐽 ∈ Top → (∃𝑦(𝑦𝐽𝑥 = 𝑦) → 𝑥𝐽))
81, 7sylbid 240 . . 3 (𝐽 ∈ Top → (𝑥 ∈ (topGen‘𝐽) → 𝑥𝐽))
98ssrdv 3988 . 2 (𝐽 ∈ Top → (topGen‘𝐽) ⊆ 𝐽)
10 bastg 22974 . 2 (𝐽 ∈ Top → 𝐽 ⊆ (topGen‘𝐽))
119, 10eqssd 4000 1 (𝐽 ∈ Top → (topGen‘𝐽) = 𝐽)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1539  wex 1778  wcel 2107  wss 3950   cuni 4906  cfv 6560  topGenctg 17483  Topctop 22900
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-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-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-top 22901
This theorem is referenced by:  eltop  22982  eltop2  22983  eltop3  22984  bastop  22989  tgtop11  22990  basgen  22996  tgfiss  22999  bastop1  23001  resttop  23169  dis1stc  23508  alexsubALTlem1  24056  xrtgioo  24829  topfne  36356  topfneec  36357  topfneec2  36358  dissneqlem  37342
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