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Theorem tgtop 20758
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 20747 . . . 4 (𝐽 ∈ Top → (𝑥 ∈ (topGen‘𝐽) ↔ ∃𝑦(𝑦𝐽𝑥 = 𝑦)))
2 simpr 477 . . . . . . 7 (((𝐽 ∈ Top ∧ 𝑦𝐽) ∧ 𝑥 = 𝑦) → 𝑥 = 𝑦)
3 uniopn 20683 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝑦𝐽) → 𝑦𝐽)
43adantr 481 . . . . . . 7 (((𝐽 ∈ Top ∧ 𝑦𝐽) ∧ 𝑥 = 𝑦) → 𝑦𝐽)
52, 4eqeltrd 2699 . . . . . 6 (((𝐽 ∈ Top ∧ 𝑦𝐽) ∧ 𝑥 = 𝑦) → 𝑥𝐽)
65expl 647 . . . . 5 (𝐽 ∈ Top → ((𝑦𝐽𝑥 = 𝑦) → 𝑥𝐽))
76exlimdv 1859 . . . 4 (𝐽 ∈ Top → (∃𝑦(𝑦𝐽𝑥 = 𝑦) → 𝑥𝐽))
81, 7sylbid 230 . . 3 (𝐽 ∈ Top → (𝑥 ∈ (topGen‘𝐽) → 𝑥𝐽))
98ssrdv 3601 . 2 (𝐽 ∈ Top → (topGen‘𝐽) ⊆ 𝐽)
10 bastg 20751 . 2 (𝐽 ∈ Top → 𝐽 ⊆ (topGen‘𝐽))
119, 10eqssd 3612 1 (𝐽 ∈ Top → (topGen‘𝐽) = 𝐽)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1481  wex 1702  wcel 1988  wss 3567   cuni 4427  cfv 5876  topGenctg 16079  Topctop 20679
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-rab 2918  df-v 3197  df-sbc 3430  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-br 4645  df-opab 4704  df-mpt 4721  df-id 5014  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-iota 5839  df-fun 5878  df-fv 5884  df-topgen 16085  df-top 20680
This theorem is referenced by:  eltop  20759  eltop2  20760  eltop3  20761  bastop  20766  tgtop11  20767  basgen  20773  tgfiss  20776  bastop1  20778  resttop  20945  dis1stc  21283  alexsubALTlem1  21832  xrtgioo  22590  topfne  32324  topfneec  32325  topfneec2  32326  dissneqlem  33158
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