Theorem bastop1 12723

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 12703	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝐵 ⊆ 𝐽) → (topGen‘𝐵) ⊆ (topGen‘𝐽))
2		tgtop 12708	. . . . . 6 ⊢ (𝐽 ∈ Top → (topGen‘𝐽) = 𝐽)
3	2	adantr 274	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝐵 ⊆ 𝐽) → (topGen‘𝐽) = 𝐽)
4	1, 3	sseqtrd 3180	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝐵 ⊆ 𝐽) → (topGen‘𝐵) ⊆ 𝐽)
5		eqss 3157	. . . . 5 ⊢ ((topGen‘𝐵) = 𝐽 ↔ ((topGen‘𝐵) ⊆ 𝐽 ∧ 𝐽 ⊆ (topGen‘𝐵)))
6	5	baib 909	. . . 4 ⊢ ((topGen‘𝐵) ⊆ 𝐽 → ((topGen‘𝐵) = 𝐽 ↔ 𝐽 ⊆ (topGen‘𝐵)))
7	4, 6	syl 14	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝐵 ⊆ 𝐽) → ((topGen‘𝐵) = 𝐽 ↔ 𝐽 ⊆ (topGen‘𝐵)))
8		dfss3 3132	. . 3 ⊢ (𝐽 ⊆ (topGen‘𝐵) ↔ ∀𝑥 ∈ 𝐽 𝑥 ∈ (topGen‘𝐵))
9	7, 8	bitrdi 195	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝐵 ⊆ 𝐽) → ((topGen‘𝐵) = 𝐽 ↔ ∀𝑥 ∈ 𝐽 𝑥 ∈ (topGen‘𝐵)))
10		ssexg 4121	. . . . 5 ⊢ ((𝐵 ⊆ 𝐽 ∧ 𝐽 ∈ Top) → 𝐵 ∈ V)
11	10	ancoms 266	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝐵 ⊆ 𝐽) → 𝐵 ∈ V)
12		eltg3 12697	. . . 4 ⊢ (𝐵 ∈ V → (𝑥 ∈ (topGen‘𝐵) ↔ ∃𝑦(𝑦 ⊆ 𝐵 ∧ 𝑥 = ∪ 𝑦)))
13	11, 12	syl 14	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝐵 ⊆ 𝐽) → (𝑥 ∈ (topGen‘𝐵) ↔ ∃𝑦(𝑦 ⊆ 𝐵 ∧ 𝑥 = ∪ 𝑦)))
14	13	ralbidv 2466	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝐵 ⊆ 𝐽) → (∀𝑥 ∈ 𝐽 𝑥 ∈ (topGen‘𝐵) ↔ ∀𝑥 ∈ 𝐽 ∃𝑦(𝑦 ⊆ 𝐵 ∧ 𝑥 = ∪ 𝑦)))
15	9, 14	bitrd 187	1 ⊢ ((𝐽 ∈ Top ∧ 𝐵 ⊆ 𝐽) → ((topGen‘𝐵) = 𝐽 ↔ ∀𝑥 ∈ 𝐽 ∃𝑦(𝑦 ⊆ 𝐵 ∧ 𝑥 = ∪ 𝑦)))

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   = wceq 1343  ∃wex 1480   ∈ wcel 2136  ∀wral 2444  Vcvv 2726   ⊆ wss 3116  ∪ cuni 3789  ‘cfv 5188  topGenctg 12571  Topctop 12635

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187  ax-un 4411

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-sbc 2952  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-br 3983  df-opab 4044  df-mpt 4045  df-id 4271  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-iota 5153  df-fun 5190  df-fv 5196  df-topgen 12577  df-top 12636

This theorem is referenced by:  bastop2  12724