Theorem bastop1 23016

Description: A subset of a topology is a basis for the topology iff every member of the topology is a union of members of the basis. We use the idiom "(topGen‘𝐵) = 𝐽 " to express "𝐵 is a basis for topology 𝐽 " since we do not have a separate notation for this. Definition 15.35 of [Schechter] p. 428. (Contributed by NM, 2-Feb-2008.) (Proof shortened by Mario Carneiro, 2-Sep-2015.)

Assertion

Ref	Expression
bastop1	⊢ ((𝐽 ∈ Top ∧ 𝐵 ⊆ 𝐽) → ((topGen‘𝐵) = 𝐽 ↔ ∀𝑥 ∈ 𝐽 ∃𝑦(𝑦 ⊆ 𝐵 ∧ 𝑥 = ∪ 𝑦)))

Distinct variable groups:   𝑥,𝑦,𝐵   𝑥,𝐽,𝑦

Proof of Theorem bastop1

Step	Hyp	Ref	Expression
1		tgss 22991	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝐵 ⊆ 𝐽) → (topGen‘𝐵) ⊆ (topGen‘𝐽))
2		tgtop 22996	. . . . . 6 ⊢ (𝐽 ∈ Top → (topGen‘𝐽) = 𝐽)
3	2	adantr 480	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝐵 ⊆ 𝐽) → (topGen‘𝐽) = 𝐽)
4	1, 3	sseqtrd 4036	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝐵 ⊆ 𝐽) → (topGen‘𝐵) ⊆ 𝐽)
5		eqss 4011	. . . . 5 ⊢ ((topGen‘𝐵) = 𝐽 ↔ ((topGen‘𝐵) ⊆ 𝐽 ∧ 𝐽 ⊆ (topGen‘𝐵)))
6	5	baib 535	. . . 4 ⊢ ((topGen‘𝐵) ⊆ 𝐽 → ((topGen‘𝐵) = 𝐽 ↔ 𝐽 ⊆ (topGen‘𝐵)))
7	4, 6	syl 17	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝐵 ⊆ 𝐽) → ((topGen‘𝐵) = 𝐽 ↔ 𝐽 ⊆ (topGen‘𝐵)))
8		dfss3 3984	. . 3 ⊢ (𝐽 ⊆ (topGen‘𝐵) ↔ ∀𝑥 ∈ 𝐽 𝑥 ∈ (topGen‘𝐵))
9	7, 8	bitrdi 287	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝐵 ⊆ 𝐽) → ((topGen‘𝐵) = 𝐽 ↔ ∀𝑥 ∈ 𝐽 𝑥 ∈ (topGen‘𝐵)))
10		ssexg 5329	. . . . 5 ⊢ ((𝐵 ⊆ 𝐽 ∧ 𝐽 ∈ Top) → 𝐵 ∈ V)
11	10	ancoms 458	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝐵 ⊆ 𝐽) → 𝐵 ∈ V)
12		eltg3 22985	. . . 4 ⊢ (𝐵 ∈ V → (𝑥 ∈ (topGen‘𝐵) ↔ ∃𝑦(𝑦 ⊆ 𝐵 ∧ 𝑥 = ∪ 𝑦)))
13	11, 12	syl 17	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝐵 ⊆ 𝐽) → (𝑥 ∈ (topGen‘𝐵) ↔ ∃𝑦(𝑦 ⊆ 𝐵 ∧ 𝑥 = ∪ 𝑦)))
14	13	ralbidv 3176	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝐵 ⊆ 𝐽) → (∀𝑥 ∈ 𝐽 𝑥 ∈ (topGen‘𝐵) ↔ ∀𝑥 ∈ 𝐽 ∃𝑦(𝑦 ⊆ 𝐵 ∧ 𝑥 = ∪ 𝑦)))
15	9, 14	bitrd 279	1 ⊢ ((𝐽 ∈ Top ∧ 𝐵 ⊆ 𝐽) → ((topGen‘𝐵) = 𝐽 ↔ ∀𝑥 ∈ 𝐽 ∃𝑦(𝑦 ⊆ 𝐵 ∧ 𝑥 = ∪ 𝑦)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1537  ∃wex 1776   ∈ wcel 2106  ∀wral 3059  Vcvv 3478   ⊆ wss 3963  ∪ cuni 4912  ‘cfv 6563  topGenctg 17484  Topctop 22915

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-iota 6516  df-fun 6565  df-fv 6571  df-topgen 17490  df-top 22916

This theorem is referenced by:  bastop2  23017