Theorem topbas 23029

Description: A topology is its own basis. (Contributed by NM, 17-Jul-2006.)

Assertion

Ref	Expression
topbas	⊢ (𝐽 ∈ Top → 𝐽 ∈ TopBases)

Proof of Theorem topbas

Dummy variables 𝑥 𝑦 𝑧 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		inopn 22956	. . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ 𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐽) → (𝑥 ∩ 𝑦) ∈ 𝐽)
2	1	3expb 1133	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐽)) → (𝑥 ∩ 𝑦) ∈ 𝐽)
3		simpr 488	. . . . . . 7 ⊢ (((𝐽 ∈ Top ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐽)) ∧ 𝑧 ∈ (𝑥 ∩ 𝑦)) → 𝑧 ∈ (𝑥 ∩ 𝑦))
4		ssid 3958	. . . . . . 7 ⊢ (𝑥 ∩ 𝑦) ⊆ (𝑥 ∩ 𝑦)
5	3, 4	jctir 528	. . . . . 6 ⊢ (((𝐽 ∈ Top ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐽)) ∧ 𝑧 ∈ (𝑥 ∩ 𝑦)) → (𝑧 ∈ (𝑥 ∩ 𝑦) ∧ (𝑥 ∩ 𝑦) ⊆ (𝑥 ∩ 𝑦)))
6		eleq2 2851	. . . . . . . 8 ⊢ (𝑤 = (𝑥 ∩ 𝑦) → (𝑧 ∈ 𝑤 ↔ 𝑧 ∈ (𝑥 ∩ 𝑦)))
7		sseq1 3961	. . . . . . . 8 ⊢ (𝑤 = (𝑥 ∩ 𝑦) → (𝑤 ⊆ (𝑥 ∩ 𝑦) ↔ (𝑥 ∩ 𝑦) ⊆ (𝑥 ∩ 𝑦)))
8	6, 7	anbi12d 641	. . . . . . 7 ⊢ (𝑤 = (𝑥 ∩ 𝑦) → ((𝑧 ∈ 𝑤 ∧ 𝑤 ⊆ (𝑥 ∩ 𝑦)) ↔ (𝑧 ∈ (𝑥 ∩ 𝑦) ∧ (𝑥 ∩ 𝑦) ⊆ (𝑥 ∩ 𝑦))))
9	8	rspcev 3581	. . . . . 6 ⊢ (((𝑥 ∩ 𝑦) ∈ 𝐽 ∧ (𝑧 ∈ (𝑥 ∩ 𝑦) ∧ (𝑥 ∩ 𝑦) ⊆ (𝑥 ∩ 𝑦))) → ∃𝑤 ∈ 𝐽 (𝑧 ∈ 𝑤 ∧ 𝑤 ⊆ (𝑥 ∩ 𝑦)))
10	2, 5, 9	syl2an2r 695	. . . . 5 ⊢ (((𝐽 ∈ Top ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐽)) ∧ 𝑧 ∈ (𝑥 ∩ 𝑦)) → ∃𝑤 ∈ 𝐽 (𝑧 ∈ 𝑤 ∧ 𝑤 ⊆ (𝑥 ∩ 𝑦)))
11	10	exp31 423	. . . 4 ⊢ (𝐽 ∈ Top → ((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐽) → (𝑧 ∈ (𝑥 ∩ 𝑦) → ∃𝑤 ∈ 𝐽 (𝑧 ∈ 𝑤 ∧ 𝑤 ⊆ (𝑥 ∩ 𝑦)))))
12	11	ralrimdv 3160	. . 3 ⊢ (𝐽 ∈ Top → ((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐽) → ∀𝑧 ∈ (𝑥 ∩ 𝑦)∃𝑤 ∈ 𝐽 (𝑧 ∈ 𝑤 ∧ 𝑤 ⊆ (𝑥 ∩ 𝑦))))
13	12	ralrimivv 3203	. 2 ⊢ (𝐽 ∈ Top → ∀𝑥 ∈ 𝐽 ∀𝑦 ∈ 𝐽 ∀𝑧 ∈ (𝑥 ∩ 𝑦)∃𝑤 ∈ 𝐽 (𝑧 ∈ 𝑤 ∧ 𝑤 ⊆ (𝑥 ∩ 𝑦)))
14		isbasis2g 23005	. 2 ⊢ (𝐽 ∈ Top → (𝐽 ∈ TopBases ↔ ∀𝑥 ∈ 𝐽 ∀𝑦 ∈ 𝐽 ∀𝑧 ∈ (𝑥 ∩ 𝑦)∃𝑤 ∈ 𝐽 (𝑧 ∈ 𝑤 ∧ 𝑤 ⊆ (𝑥 ∩ 𝑦))))
15	13, 14	mpbird 259	1 ⊢ (𝐽 ∈ Top → 𝐽 ∈ TopBases)

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1560   ∈ wcel 2142  ∀wral 3076  ∃wrex 3086   ∩ cin 3903   ⊆ wss 3904  Topctop 22950  TopBasesctb 23002

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-ext 2734  ax-sep 5246

This theorem depends on definitions:  df-bi 209  df-an 400  df-3an 1100  df-tru 1563  df-ex 1800  df-sb 2091  df-clab 2741  df-cleq 2754  df-clel 2837  df-ral 3077  df-rex 3087  df-rab 3415  df-v 3456  df-in 3911  df-ss 3921  df-pw 4557  df-uni 4866  df-top 22951  df-bases 23003

This theorem is referenced by:  resttop  23217  dis1stc  23556  txtop  23626  onpsstopbas  36787