Theorem topbas 23299

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 23226	. . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ 𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐽) → (𝑥 ∩ 𝑦) ∈ 𝐽)
2	1	3expb 1138	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐽)) → (𝑥 ∩ 𝑦) ∈ 𝐽)
3		simpr 490	. . . . . . 7 ⊢ (((𝐽 ∈ Top ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐽)) ∧ 𝑧 ∈ (𝑥 ∩ 𝑦)) → 𝑧 ∈ (𝑥 ∩ 𝑦))
4		ssid 3931	. . . . . . 7 ⊢ (𝑥 ∩ 𝑦) ⊆ (𝑥 ∩ 𝑦)
5	3, 4	jctir 530	. . . . . 6 ⊢ (((𝐽 ∈ Top ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐽)) ∧ 𝑧 ∈ (𝑥 ∩ 𝑦)) → (𝑧 ∈ (𝑥 ∩ 𝑦) ∧ (𝑥 ∩ 𝑦) ⊆ (𝑥 ∩ 𝑦)))
6		eleq2 2849	. . . . . . . 8 ⊢ (𝑤 = (𝑥 ∩ 𝑦) → (𝑧 ∈ 𝑤 ↔ 𝑧 ∈ (𝑥 ∩ 𝑦)))
7		sseq1 3934	. . . . . . . 8 ⊢ (𝑤 = (𝑥 ∩ 𝑦) → (𝑤 ⊆ (𝑥 ∩ 𝑦) ↔ (𝑥 ∩ 𝑦) ⊆ (𝑥 ∩ 𝑦)))
8	6, 7	anbi12d 644	. . . . . . 7 ⊢ (𝑤 = (𝑥 ∩ 𝑦) → ((𝑧 ∈ 𝑤 ∧ 𝑤 ⊆ (𝑥 ∩ 𝑦)) ↔ (𝑧 ∈ (𝑥 ∩ 𝑦) ∧ (𝑥 ∩ 𝑦) ⊆ (𝑥 ∩ 𝑦))))
9	8	rspcev 3576	. . . . . 6 ⊢ (((𝑥 ∩ 𝑦) ∈ 𝐽 ∧ (𝑧 ∈ (𝑥 ∩ 𝑦) ∧ (𝑥 ∩ 𝑦) ⊆ (𝑥 ∩ 𝑦))) → ∃𝑤 ∈ 𝐽 (𝑧 ∈ 𝑤 ∧ 𝑤 ⊆ (𝑥 ∩ 𝑦)))
10	2, 5, 9	syl2an2r 698	. . . . 5 ⊢ (((𝐽 ∈ Top ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐽)) ∧ 𝑧 ∈ (𝑥 ∩ 𝑦)) → ∃𝑤 ∈ 𝐽 (𝑧 ∈ 𝑤 ∧ 𝑤 ⊆ (𝑥 ∩ 𝑦)))
11	10	exp31 425	. . . 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 23275	. 2 ⊢ (𝐽 ∈ Top → (𝐽 ∈ TopBases ↔ ∀𝑥 ∈ 𝐽 ∀𝑦 ∈ 𝐽 ∀𝑧 ∈ (𝑥 ∩ 𝑦)∃𝑤 ∈ 𝐽 (𝑧 ∈ 𝑤 ∧ 𝑤 ⊆ (𝑥 ∩ 𝑦))))
15	13, 14	mpbird 260	1 ⊢ (𝐽 ∈ Top → 𝐽 ∈ TopBases)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1569   ∈ wcel 2144  ∀wral 3076  ∃wrex 3086   ⊆ wss 3896   ∩ cin 4088  Topctop 23220  TopBasesctb 23272

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732  ax-sep 5248

This proof depends on definitions:  df-bi 210  df-an 402  df-3an 1105  df-tru 1572  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3897  df-in 4090  df-pw 4558  df-uni 4867  df-top 23221  df-bases 23273

This theorem is used by:  resttop  23487  dis1stc  23827  txtop  23897  onpsstopbas  37342