Theorem isbasis2g 14897

Description: Express the predicate "the set 𝐵 is a basis for a topology". (Contributed by NM, 17-Jul-2006.)

Assertion

Ref	Expression
isbasis2g	⊢ (𝐵 ∈ 𝐶 → (𝐵 ∈ TopBases ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ (𝑥 ∩ 𝑦)∃𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 ∧ 𝑤 ⊆ (𝑥 ∩ 𝑦))))

Distinct variable group:   𝑥,𝑤,𝑦,𝑧,𝐵

Allowed substitution hints:   𝐶(𝑥,𝑦,𝑧,𝑤)

Proof of Theorem isbasis2g

Step	Hyp	Ref	Expression
1		isbasisg 14896	. 2 ⊢ (𝐵 ∈ 𝐶 → (𝐵 ∈ TopBases ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 ∩ 𝑦) ⊆ ∪ (𝐵 ∩ 𝒫 (𝑥 ∩ 𝑦))))
2		dfss3 3226	. . . 4 ⊢ ((𝑥 ∩ 𝑦) ⊆ ∪ (𝐵 ∩ 𝒫 (𝑥 ∩ 𝑦)) ↔ ∀𝑧 ∈ (𝑥 ∩ 𝑦)𝑧 ∈ ∪ (𝐵 ∩ 𝒫 (𝑥 ∩ 𝑦)))
3		elin 3401	. . . . . . . . . 10 ⊢ (𝑤 ∈ (𝐵 ∩ 𝒫 (𝑥 ∩ 𝑦)) ↔ (𝑤 ∈ 𝐵 ∧ 𝑤 ∈ 𝒫 (𝑥 ∩ 𝑦)))
4		velpw 3675	. . . . . . . . . . 11 ⊢ (𝑤 ∈ 𝒫 (𝑥 ∩ 𝑦) ↔ 𝑤 ⊆ (𝑥 ∩ 𝑦))
5	4	anbi2i 457	. . . . . . . . . 10 ⊢ ((𝑤 ∈ 𝐵 ∧ 𝑤 ∈ 𝒫 (𝑥 ∩ 𝑦)) ↔ (𝑤 ∈ 𝐵 ∧ 𝑤 ⊆ (𝑥 ∩ 𝑦)))
6	3, 5	bitri 184	. . . . . . . . 9 ⊢ (𝑤 ∈ (𝐵 ∩ 𝒫 (𝑥 ∩ 𝑦)) ↔ (𝑤 ∈ 𝐵 ∧ 𝑤 ⊆ (𝑥 ∩ 𝑦)))
7	6	anbi2i 457	. . . . . . . 8 ⊢ ((𝑧 ∈ 𝑤 ∧ 𝑤 ∈ (𝐵 ∩ 𝒫 (𝑥 ∩ 𝑦))) ↔ (𝑧 ∈ 𝑤 ∧ (𝑤 ∈ 𝐵 ∧ 𝑤 ⊆ (𝑥 ∩ 𝑦))))
8		an12 563	. . . . . . . 8 ⊢ ((𝑧 ∈ 𝑤 ∧ (𝑤 ∈ 𝐵 ∧ 𝑤 ⊆ (𝑥 ∩ 𝑦))) ↔ (𝑤 ∈ 𝐵 ∧ (𝑧 ∈ 𝑤 ∧ 𝑤 ⊆ (𝑥 ∩ 𝑦))))
9	7, 8	bitri 184	. . . . . . 7 ⊢ ((𝑧 ∈ 𝑤 ∧ 𝑤 ∈ (𝐵 ∩ 𝒫 (𝑥 ∩ 𝑦))) ↔ (𝑤 ∈ 𝐵 ∧ (𝑧 ∈ 𝑤 ∧ 𝑤 ⊆ (𝑥 ∩ 𝑦))))
10	9	exbii 1654	. . . . . 6 ⊢ (∃𝑤(𝑧 ∈ 𝑤 ∧ 𝑤 ∈ (𝐵 ∩ 𝒫 (𝑥 ∩ 𝑦))) ↔ ∃𝑤(𝑤 ∈ 𝐵 ∧ (𝑧 ∈ 𝑤 ∧ 𝑤 ⊆ (𝑥 ∩ 𝑦))))
11		eluni 3916	. . . . . 6 ⊢ (𝑧 ∈ ∪ (𝐵 ∩ 𝒫 (𝑥 ∩ 𝑦)) ↔ ∃𝑤(𝑧 ∈ 𝑤 ∧ 𝑤 ∈ (𝐵 ∩ 𝒫 (𝑥 ∩ 𝑦))))
12		df-rex 2526	. . . . . 6 ⊢ (∃𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 ∧ 𝑤 ⊆ (𝑥 ∩ 𝑦)) ↔ ∃𝑤(𝑤 ∈ 𝐵 ∧ (𝑧 ∈ 𝑤 ∧ 𝑤 ⊆ (𝑥 ∩ 𝑦))))
13	10, 11, 12	3bitr4i 212	. . . . 5 ⊢ (𝑧 ∈ ∪ (𝐵 ∩ 𝒫 (𝑥 ∩ 𝑦)) ↔ ∃𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 ∧ 𝑤 ⊆ (𝑥 ∩ 𝑦)))
14	13	ralbii 2548	. . . 4 ⊢ (∀𝑧 ∈ (𝑥 ∩ 𝑦)𝑧 ∈ ∪ (𝐵 ∩ 𝒫 (𝑥 ∩ 𝑦)) ↔ ∀𝑧 ∈ (𝑥 ∩ 𝑦)∃𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 ∧ 𝑤 ⊆ (𝑥 ∩ 𝑦)))
15	2, 14	bitri 184	. . 3 ⊢ ((𝑥 ∩ 𝑦) ⊆ ∪ (𝐵 ∩ 𝒫 (𝑥 ∩ 𝑦)) ↔ ∀𝑧 ∈ (𝑥 ∩ 𝑦)∃𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 ∧ 𝑤 ⊆ (𝑥 ∩ 𝑦)))
16	15	2ralbii 2550	. 2 ⊢ (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 ∩ 𝑦) ⊆ ∪ (𝐵 ∩ 𝒫 (𝑥 ∩ 𝑦)) ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ (𝑥 ∩ 𝑦)∃𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 ∧ 𝑤 ⊆ (𝑥 ∩ 𝑦)))
17	1, 16	bitrdi 196	1 ⊢ (𝐵 ∈ 𝐶 → (𝐵 ∈ TopBases ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ (𝑥 ∩ 𝑦)∃𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 ∧ 𝑤 ⊆ (𝑥 ∩ 𝑦))))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105  ∃wex 1541   ∈ wcel 2203  ∀wral 2520  ∃wrex 2521   ∩ cin 3209   ⊆ wss 3210  𝒫 cpw 3668  ∪ cuni 3913  TopBasesctb 14894

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2214

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-v 2814  df-in 3216  df-ss 3223  df-pw 3670  df-uni 3914  df-bases 14895

This theorem is referenced by:  isbasis3g  14898  basis2  14900  fiinbas  14901  tgclb  14917  topbas  14919  restbasg  15020  txbas  15110  blbas  15285