Theorem bastop1 20845

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 20820	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝐵 ⊆ 𝐽) → (topGen‘𝐵) ⊆ (topGen‘𝐽))
2		tgtop 20825	. . . . . 6 ⊢ (𝐽 ∈ Top → (topGen‘𝐽) = 𝐽)
3	2	adantr 480	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝐵 ⊆ 𝐽) → (topGen‘𝐽) = 𝐽)
4	1, 3	sseqtrd 3674	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝐵 ⊆ 𝐽) → (topGen‘𝐵) ⊆ 𝐽)
5		eqss 3651	. . . . 5 ⊢ ((topGen‘𝐵) = 𝐽 ↔ ((topGen‘𝐵) ⊆ 𝐽 ∧ 𝐽 ⊆ (topGen‘𝐵)))
6	5	baib 964	. . . 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 3625	. . 3 ⊢ (𝐽 ⊆ (topGen‘𝐵) ↔ ∀𝑥 ∈ 𝐽 𝑥 ∈ (topGen‘𝐵))
9	7, 8	syl6bb 276	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝐵 ⊆ 𝐽) → ((topGen‘𝐵) = 𝐽 ↔ ∀𝑥 ∈ 𝐽 𝑥 ∈ (topGen‘𝐵)))
10		ssexg 4837	. . . . 5 ⊢ ((𝐵 ⊆ 𝐽 ∧ 𝐽 ∈ Top) → 𝐵 ∈ V)
11	10	ancoms 468	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝐵 ⊆ 𝐽) → 𝐵 ∈ V)
12		eltg3 20814	. . . 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 3015	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝐵 ⊆ 𝐽) → (∀𝑥 ∈ 𝐽 𝑥 ∈ (topGen‘𝐵) ↔ ∀𝑥 ∈ 𝐽 ∃𝑦(𝑦 ⊆ 𝐵 ∧ 𝑥 = ∪ 𝑦)))
15	9, 14	bitrd 268	1 ⊢ ((𝐽 ∈ Top ∧ 𝐵 ⊆ 𝐽) → ((topGen‘𝐵) = 𝐽 ↔ ∀𝑥 ∈ 𝐽 ∃𝑦(𝑦 ⊆ 𝐵 ∧ 𝑥 = ∪ 𝑦)))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 383   = wceq 1523  ∃wex 1744   ∈ wcel 2030  ∀wral 2941  Vcvv 3231   ⊆ wss 3607  ∪ cuni 4468  ‘cfv 5926  topGenctg 16145  Topctop 20746

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-iota 5889  df-fun 5928  df-fv 5934  df-topgen 16151  df-top 20747

This theorem is referenced by:  bastop2  20846