Theorem bastop1 13365

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 13345	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝐵 ⊆ 𝐽) → (topGen‘𝐵) ⊆ (topGen‘𝐽))
2		tgtop 13350	. . . . . 6 ⊢ (𝐽 ∈ Top → (topGen‘𝐽) = 𝐽)
3	2	adantr 276	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝐵 ⊆ 𝐽) → (topGen‘𝐽) = 𝐽)
4	1, 3	sseqtrd 3193	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝐵 ⊆ 𝐽) → (topGen‘𝐵) ⊆ 𝐽)
5		eqss 3170	. . . . 5 ⊢ ((topGen‘𝐵) = 𝐽 ↔ ((topGen‘𝐵) ⊆ 𝐽 ∧ 𝐽 ⊆ (topGen‘𝐵)))
6	5	baib 919	. . . 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 3145	. . 3 ⊢ (𝐽 ⊆ (topGen‘𝐵) ↔ ∀𝑥 ∈ 𝐽 𝑥 ∈ (topGen‘𝐵))
9	7, 8	bitrdi 196	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝐵 ⊆ 𝐽) → ((topGen‘𝐵) = 𝐽 ↔ ∀𝑥 ∈ 𝐽 𝑥 ∈ (topGen‘𝐵)))
10		ssexg 4140	. . . . 5 ⊢ ((𝐵 ⊆ 𝐽 ∧ 𝐽 ∈ Top) → 𝐵 ∈ V)
11	10	ancoms 268	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝐵 ⊆ 𝐽) → 𝐵 ∈ V)
12		eltg3 13339	. . . 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 2477	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝐵 ⊆ 𝐽) → (∀𝑥 ∈ 𝐽 𝑥 ∈ (topGen‘𝐵) ↔ ∀𝑥 ∈ 𝐽 ∃𝑦(𝑦 ⊆ 𝐵 ∧ 𝑥 = ∪ 𝑦)))
15	9, 14	bitrd 188	1 ⊢ ((𝐽 ∈ Top ∧ 𝐵 ⊆ 𝐽) → ((topGen‘𝐵) = 𝐽 ↔ ∀𝑥 ∈ 𝐽 ∃𝑦(𝑦 ⊆ 𝐵 ∧ 𝑥 = ∪ 𝑦)))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1353  ∃wex 1492   ∈ wcel 2148  ∀wral 2455  Vcvv 2737   ⊆ wss 3129  ∪ cuni 3808  ‘cfv 5213  topGenctg 12689  Topctop 13277

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4119  ax-pow 4172  ax-pr 4207  ax-un 4431

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-sbc 2963  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3809  df-br 4002  df-opab 4063  df-mpt 4064  df-id 4291  df-xp 4630  df-rel 4631  df-cnv 4632  df-co 4633  df-dm 4634  df-iota 5175  df-fun 5215  df-fv 5221  df-topgen 12695  df-top 13278

This theorem is referenced by:  bastop2  13366