Theorem topbas 22475

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 22401	. . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ 𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐽) → (𝑥 ∩ 𝑦) ∈ 𝐽)
2	1	3expb 1121	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐽)) → (𝑥 ∩ 𝑦) ∈ 𝐽)
3		simpr 486	. . . . . . 7 ⊢ (((𝐽 ∈ Top ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐽)) ∧ 𝑧 ∈ (𝑥 ∩ 𝑦)) → 𝑧 ∈ (𝑥 ∩ 𝑦))
4		ssid 4005	. . . . . . 7 ⊢ (𝑥 ∩ 𝑦) ⊆ (𝑥 ∩ 𝑦)
5	3, 4	jctir 522	. . . . . 6 ⊢ (((𝐽 ∈ Top ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐽)) ∧ 𝑧 ∈ (𝑥 ∩ 𝑦)) → (𝑧 ∈ (𝑥 ∩ 𝑦) ∧ (𝑥 ∩ 𝑦) ⊆ (𝑥 ∩ 𝑦)))
6		eleq2 2823	. . . . . . . 8 ⊢ (𝑤 = (𝑥 ∩ 𝑦) → (𝑧 ∈ 𝑤 ↔ 𝑧 ∈ (𝑥 ∩ 𝑦)))
7		sseq1 4008	. . . . . . . 8 ⊢ (𝑤 = (𝑥 ∩ 𝑦) → (𝑤 ⊆ (𝑥 ∩ 𝑦) ↔ (𝑥 ∩ 𝑦) ⊆ (𝑥 ∩ 𝑦)))
8	6, 7	anbi12d 632	. . . . . . 7 ⊢ (𝑤 = (𝑥 ∩ 𝑦) → ((𝑧 ∈ 𝑤 ∧ 𝑤 ⊆ (𝑥 ∩ 𝑦)) ↔ (𝑧 ∈ (𝑥 ∩ 𝑦) ∧ (𝑥 ∩ 𝑦) ⊆ (𝑥 ∩ 𝑦))))
9	8	rspcev 3613	. . . . . 6 ⊢ (((𝑥 ∩ 𝑦) ∈ 𝐽 ∧ (𝑧 ∈ (𝑥 ∩ 𝑦) ∧ (𝑥 ∩ 𝑦) ⊆ (𝑥 ∩ 𝑦))) → ∃𝑤 ∈ 𝐽 (𝑧 ∈ 𝑤 ∧ 𝑤 ⊆ (𝑥 ∩ 𝑦)))
10	2, 5, 9	syl2an2r 684	. . . . 5 ⊢ (((𝐽 ∈ Top ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐽)) ∧ 𝑧 ∈ (𝑥 ∩ 𝑦)) → ∃𝑤 ∈ 𝐽 (𝑧 ∈ 𝑤 ∧ 𝑤 ⊆ (𝑥 ∩ 𝑦)))
11	10	exp31 421	. . . 4 ⊢ (𝐽 ∈ Top → ((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐽) → (𝑧 ∈ (𝑥 ∩ 𝑦) → ∃𝑤 ∈ 𝐽 (𝑧 ∈ 𝑤 ∧ 𝑤 ⊆ (𝑥 ∩ 𝑦)))))
12	11	ralrimdv 3153	. . 3 ⊢ (𝐽 ∈ Top → ((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐽) → ∀𝑧 ∈ (𝑥 ∩ 𝑦)∃𝑤 ∈ 𝐽 (𝑧 ∈ 𝑤 ∧ 𝑤 ⊆ (𝑥 ∩ 𝑦))))
13	12	ralrimivv 3199	. 2 ⊢ (𝐽 ∈ Top → ∀𝑥 ∈ 𝐽 ∀𝑦 ∈ 𝐽 ∀𝑧 ∈ (𝑥 ∩ 𝑦)∃𝑤 ∈ 𝐽 (𝑧 ∈ 𝑤 ∧ 𝑤 ⊆ (𝑥 ∩ 𝑦)))
14		isbasis2g 22451	. 2 ⊢ (𝐽 ∈ Top → (𝐽 ∈ TopBases ↔ ∀𝑥 ∈ 𝐽 ∀𝑦 ∈ 𝐽 ∀𝑧 ∈ (𝑥 ∩ 𝑦)∃𝑤 ∈ 𝐽 (𝑧 ∈ 𝑤 ∧ 𝑤 ⊆ (𝑥 ∩ 𝑦))))
15	13, 14	mpbird 257	1 ⊢ (𝐽 ∈ Top → 𝐽 ∈ TopBases)

Syntax hints:   → wi 4   ∧ wa 397   = wceq 1542   ∈ wcel 2107  ∀wral 3062  ∃wrex 3071   ∩ cin 3948   ⊆ wss 3949  Topctop 22395  TopBasesctb 22448

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5300

This theorem depends on definitions:  df-bi 206  df-an 398  df-3an 1090  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-in 3956  df-ss 3966  df-pw 4605  df-uni 4910  df-top 22396  df-bases 22449

This theorem is referenced by:  resttop  22664  dis1stc  23003  txtop  23073  onpsstopbas  35315