Theorem basis2 11913

Description: Property of a basis. (Contributed by NM, 17-Jul-2006.)

Assertion

Ref	Expression
basis2	⊢ (((𝐵 ∈ TopBases ∧ 𝐶 ∈ 𝐵) ∧ (𝐷 ∈ 𝐵 ∧ 𝐴 ∈ (𝐶 ∩ 𝐷))) → ∃𝑥 ∈ 𝐵 (𝐴 ∈ 𝑥 ∧ 𝑥 ⊆ (𝐶 ∩ 𝐷)))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝐶   𝑥,𝐷

Proof of Theorem basis2

Dummy variables 𝑤 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		isbasis2g 11910	. . . . 5 ⊢ (𝐵 ∈ TopBases → (𝐵 ∈ TopBases ↔ ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ (𝑦 ∩ 𝑧)∃𝑥 ∈ 𝐵 (𝑤 ∈ 𝑥 ∧ 𝑥 ⊆ (𝑦 ∩ 𝑧))))
2	1	ibi 175	. . . 4 ⊢ (𝐵 ∈ TopBases → ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ (𝑦 ∩ 𝑧)∃𝑥 ∈ 𝐵 (𝑤 ∈ 𝑥 ∧ 𝑥 ⊆ (𝑦 ∩ 𝑧)))
3		ineq1 3209	. . . . . . 7 ⊢ (𝑦 = 𝐶 → (𝑦 ∩ 𝑧) = (𝐶 ∩ 𝑧))
4		sseq2 3063	. . . . . . . . . 10 ⊢ ((𝑦 ∩ 𝑧) = (𝐶 ∩ 𝑧) → (𝑥 ⊆ (𝑦 ∩ 𝑧) ↔ 𝑥 ⊆ (𝐶 ∩ 𝑧)))
5	4	anbi2d 453	. . . . . . . . 9 ⊢ ((𝑦 ∩ 𝑧) = (𝐶 ∩ 𝑧) → ((𝑤 ∈ 𝑥 ∧ 𝑥 ⊆ (𝑦 ∩ 𝑧)) ↔ (𝑤 ∈ 𝑥 ∧ 𝑥 ⊆ (𝐶 ∩ 𝑧))))
6	5	rexbidv 2392	. . . . . . . 8 ⊢ ((𝑦 ∩ 𝑧) = (𝐶 ∩ 𝑧) → (∃𝑥 ∈ 𝐵 (𝑤 ∈ 𝑥 ∧ 𝑥 ⊆ (𝑦 ∩ 𝑧)) ↔ ∃𝑥 ∈ 𝐵 (𝑤 ∈ 𝑥 ∧ 𝑥 ⊆ (𝐶 ∩ 𝑧))))
7	6	raleqbi1dv 2584	. . . . . . 7 ⊢ ((𝑦 ∩ 𝑧) = (𝐶 ∩ 𝑧) → (∀𝑤 ∈ (𝑦 ∩ 𝑧)∃𝑥 ∈ 𝐵 (𝑤 ∈ 𝑥 ∧ 𝑥 ⊆ (𝑦 ∩ 𝑧)) ↔ ∀𝑤 ∈ (𝐶 ∩ 𝑧)∃𝑥 ∈ 𝐵 (𝑤 ∈ 𝑥 ∧ 𝑥 ⊆ (𝐶 ∩ 𝑧))))
8	3, 7	syl 14	. . . . . 6 ⊢ (𝑦 = 𝐶 → (∀𝑤 ∈ (𝑦 ∩ 𝑧)∃𝑥 ∈ 𝐵 (𝑤 ∈ 𝑥 ∧ 𝑥 ⊆ (𝑦 ∩ 𝑧)) ↔ ∀𝑤 ∈ (𝐶 ∩ 𝑧)∃𝑥 ∈ 𝐵 (𝑤 ∈ 𝑥 ∧ 𝑥 ⊆ (𝐶 ∩ 𝑧))))
9		ineq2 3210	. . . . . . 7 ⊢ (𝑧 = 𝐷 → (𝐶 ∩ 𝑧) = (𝐶 ∩ 𝐷))
10		sseq2 3063	. . . . . . . . . 10 ⊢ ((𝐶 ∩ 𝑧) = (𝐶 ∩ 𝐷) → (𝑥 ⊆ (𝐶 ∩ 𝑧) ↔ 𝑥 ⊆ (𝐶 ∩ 𝐷)))
11	10	anbi2d 453	. . . . . . . . 9 ⊢ ((𝐶 ∩ 𝑧) = (𝐶 ∩ 𝐷) → ((𝑤 ∈ 𝑥 ∧ 𝑥 ⊆ (𝐶 ∩ 𝑧)) ↔ (𝑤 ∈ 𝑥 ∧ 𝑥 ⊆ (𝐶 ∩ 𝐷))))
12	11	rexbidv 2392	. . . . . . . 8 ⊢ ((𝐶 ∩ 𝑧) = (𝐶 ∩ 𝐷) → (∃𝑥 ∈ 𝐵 (𝑤 ∈ 𝑥 ∧ 𝑥 ⊆ (𝐶 ∩ 𝑧)) ↔ ∃𝑥 ∈ 𝐵 (𝑤 ∈ 𝑥 ∧ 𝑥 ⊆ (𝐶 ∩ 𝐷))))
13	12	raleqbi1dv 2584	. . . . . . 7 ⊢ ((𝐶 ∩ 𝑧) = (𝐶 ∩ 𝐷) → (∀𝑤 ∈ (𝐶 ∩ 𝑧)∃𝑥 ∈ 𝐵 (𝑤 ∈ 𝑥 ∧ 𝑥 ⊆ (𝐶 ∩ 𝑧)) ↔ ∀𝑤 ∈ (𝐶 ∩ 𝐷)∃𝑥 ∈ 𝐵 (𝑤 ∈ 𝑥 ∧ 𝑥 ⊆ (𝐶 ∩ 𝐷))))
14	9, 13	syl 14	. . . . . 6 ⊢ (𝑧 = 𝐷 → (∀𝑤 ∈ (𝐶 ∩ 𝑧)∃𝑥 ∈ 𝐵 (𝑤 ∈ 𝑥 ∧ 𝑥 ⊆ (𝐶 ∩ 𝑧)) ↔ ∀𝑤 ∈ (𝐶 ∩ 𝐷)∃𝑥 ∈ 𝐵 (𝑤 ∈ 𝑥 ∧ 𝑥 ⊆ (𝐶 ∩ 𝐷))))
15	8, 14	rspc2v 2748	. . . . 5 ⊢ ((𝐶 ∈ 𝐵 ∧ 𝐷 ∈ 𝐵) → (∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ (𝑦 ∩ 𝑧)∃𝑥 ∈ 𝐵 (𝑤 ∈ 𝑥 ∧ 𝑥 ⊆ (𝑦 ∩ 𝑧)) → ∀𝑤 ∈ (𝐶 ∩ 𝐷)∃𝑥 ∈ 𝐵 (𝑤 ∈ 𝑥 ∧ 𝑥 ⊆ (𝐶 ∩ 𝐷))))
16		eleq1 2157	. . . . . . . 8 ⊢ (𝑤 = 𝐴 → (𝑤 ∈ 𝑥 ↔ 𝐴 ∈ 𝑥))
17	16	anbi1d 454	. . . . . . 7 ⊢ (𝑤 = 𝐴 → ((𝑤 ∈ 𝑥 ∧ 𝑥 ⊆ (𝐶 ∩ 𝐷)) ↔ (𝐴 ∈ 𝑥 ∧ 𝑥 ⊆ (𝐶 ∩ 𝐷))))
18	17	rexbidv 2392	. . . . . 6 ⊢ (𝑤 = 𝐴 → (∃𝑥 ∈ 𝐵 (𝑤 ∈ 𝑥 ∧ 𝑥 ⊆ (𝐶 ∩ 𝐷)) ↔ ∃𝑥 ∈ 𝐵 (𝐴 ∈ 𝑥 ∧ 𝑥 ⊆ (𝐶 ∩ 𝐷))))
19	18	rspccv 2733	. . . . 5 ⊢ (∀𝑤 ∈ (𝐶 ∩ 𝐷)∃𝑥 ∈ 𝐵 (𝑤 ∈ 𝑥 ∧ 𝑥 ⊆ (𝐶 ∩ 𝐷)) → (𝐴 ∈ (𝐶 ∩ 𝐷) → ∃𝑥 ∈ 𝐵 (𝐴 ∈ 𝑥 ∧ 𝑥 ⊆ (𝐶 ∩ 𝐷))))
20	15, 19	syl6com 35	. . . 4 ⊢ (∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ (𝑦 ∩ 𝑧)∃𝑥 ∈ 𝐵 (𝑤 ∈ 𝑥 ∧ 𝑥 ⊆ (𝑦 ∩ 𝑧)) → ((𝐶 ∈ 𝐵 ∧ 𝐷 ∈ 𝐵) → (𝐴 ∈ (𝐶 ∩ 𝐷) → ∃𝑥 ∈ 𝐵 (𝐴 ∈ 𝑥 ∧ 𝑥 ⊆ (𝐶 ∩ 𝐷)))))
21	2, 20	syl 14	. . 3 ⊢ (𝐵 ∈ TopBases → ((𝐶 ∈ 𝐵 ∧ 𝐷 ∈ 𝐵) → (𝐴 ∈ (𝐶 ∩ 𝐷) → ∃𝑥 ∈ 𝐵 (𝐴 ∈ 𝑥 ∧ 𝑥 ⊆ (𝐶 ∩ 𝐷)))))
22	21	expd 255	. 2 ⊢ (𝐵 ∈ TopBases → (𝐶 ∈ 𝐵 → (𝐷 ∈ 𝐵 → (𝐴 ∈ (𝐶 ∩ 𝐷) → ∃𝑥 ∈ 𝐵 (𝐴 ∈ 𝑥 ∧ 𝑥 ⊆ (𝐶 ∩ 𝐷))))))
23	22	imp43 348	1 ⊢ (((𝐵 ∈ TopBases ∧ 𝐶 ∈ 𝐵) ∧ (𝐷 ∈ 𝐵 ∧ 𝐴 ∈ (𝐶 ∩ 𝐷))) → ∃𝑥 ∈ 𝐵 (𝐴 ∈ 𝑥 ∧ 𝑥 ⊆ (𝐶 ∩ 𝐷)))

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   = wceq 1296   ∈ wcel 1445  ∀wral 2370  ∃wrex 2371   ∩ cin 3012   ⊆ wss 3013  TopBasesctb 11907

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077

This theorem depends on definitions:  df-bi 116  df-tru 1299  df-nf 1402  df-sb 1700  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ral 2375  df-rex 2376  df-v 2635  df-in 3019  df-ss 3026  df-pw 3451  df-uni 3676  df-bases 11908

This theorem is referenced by:  tgcl  11931  restbasg  12035  tgioo  12335