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Theorem tgtop 21103
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 21092 . . . 4 (𝐽 ∈ Top → (𝑥 ∈ (topGen‘𝐽) ↔ ∃𝑦(𝑦𝐽𝑥 = 𝑦)))
2 simpr 478 . . . . . . 7 (((𝐽 ∈ Top ∧ 𝑦𝐽) ∧ 𝑥 = 𝑦) → 𝑥 = 𝑦)
3 uniopn 21027 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝑦𝐽) → 𝑦𝐽)
43adantr 473 . . . . . . 7 (((𝐽 ∈ Top ∧ 𝑦𝐽) ∧ 𝑥 = 𝑦) → 𝑦𝐽)
52, 4eqeltrd 2876 . . . . . 6 (((𝐽 ∈ Top ∧ 𝑦𝐽) ∧ 𝑥 = 𝑦) → 𝑥𝐽)
65expl 450 . . . . 5 (𝐽 ∈ Top → ((𝑦𝐽𝑥 = 𝑦) → 𝑥𝐽))
76exlimdv 2029 . . . 4 (𝐽 ∈ Top → (∃𝑦(𝑦𝐽𝑥 = 𝑦) → 𝑥𝐽))
81, 7sylbid 232 . . 3 (𝐽 ∈ Top → (𝑥 ∈ (topGen‘𝐽) → 𝑥𝐽))
98ssrdv 3802 . 2 (𝐽 ∈ Top → (topGen‘𝐽) ⊆ 𝐽)
10 bastg 21096 . 2 (𝐽 ∈ Top → 𝐽 ⊆ (topGen‘𝐽))
119, 10eqssd 3813 1 (𝐽 ∈ Top → (topGen‘𝐽) = 𝐽)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 385   = wceq 1653  wex 1875  wcel 2157  wss 3767   cuni 4626  cfv 6099  topGenctg 16410  Topctop 21023
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2354  ax-ext 2775  ax-sep 4973  ax-nul 4981  ax-pow 5033  ax-pr 5095  ax-un 7181
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2590  df-eu 2607  df-clab 2784  df-cleq 2790  df-clel 2793  df-nfc 2928  df-ral 3092  df-rex 3093  df-rab 3096  df-v 3385  df-sbc 3632  df-dif 3770  df-un 3772  df-in 3774  df-ss 3781  df-nul 4114  df-if 4276  df-pw 4349  df-sn 4367  df-pr 4369  df-op 4373  df-uni 4627  df-br 4842  df-opab 4904  df-mpt 4921  df-id 5218  df-xp 5316  df-rel 5317  df-cnv 5318  df-co 5319  df-dm 5320  df-iota 6062  df-fun 6101  df-fv 6107  df-topgen 16416  df-top 21024
This theorem is referenced by:  eltop  21104  eltop2  21105  eltop3  21106  bastop  21111  tgtop11  21112  basgen  21118  tgfiss  21121  bastop1  21123  resttop  21290  dis1stc  21628  alexsubALTlem1  22176  xrtgioo  22934  topfne  32853  topfneec  32854  topfneec2  32855  dissneqlem  33678
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