Theorem basis2 13931

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 13928	. . . . 5 ⊢ (𝐵 ∈ TopBases → (𝐵 ∈ TopBases ↔ ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ (𝑦 ∩ 𝑧)∃𝑥 ∈ 𝐵 (𝑤 ∈ 𝑥 ∧ 𝑥 ⊆ (𝑦 ∩ 𝑧))))
2	1	ibi 176	. . . 4 ⊢ (𝐵 ∈ TopBases → ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ (𝑦 ∩ 𝑧)∃𝑥 ∈ 𝐵 (𝑤 ∈ 𝑥 ∧ 𝑥 ⊆ (𝑦 ∩ 𝑧)))
3		ineq1 3343	. . . . . . 7 ⊢ (𝑦 = 𝐶 → (𝑦 ∩ 𝑧) = (𝐶 ∩ 𝑧))
4		sseq2 3193	. . . . . . . . . 10 ⊢ ((𝑦 ∩ 𝑧) = (𝐶 ∩ 𝑧) → (𝑥 ⊆ (𝑦 ∩ 𝑧) ↔ 𝑥 ⊆ (𝐶 ∩ 𝑧)))
5	4	anbi2d 464	. . . . . . . . 9 ⊢ ((𝑦 ∩ 𝑧) = (𝐶 ∩ 𝑧) → ((𝑤 ∈ 𝑥 ∧ 𝑥 ⊆ (𝑦 ∩ 𝑧)) ↔ (𝑤 ∈ 𝑥 ∧ 𝑥 ⊆ (𝐶 ∩ 𝑧))))
6	5	rexbidv 2490	. . . . . . . 8 ⊢ ((𝑦 ∩ 𝑧) = (𝐶 ∩ 𝑧) → (∃𝑥 ∈ 𝐵 (𝑤 ∈ 𝑥 ∧ 𝑥 ⊆ (𝑦 ∩ 𝑧)) ↔ ∃𝑥 ∈ 𝐵 (𝑤 ∈ 𝑥 ∧ 𝑥 ⊆ (𝐶 ∩ 𝑧))))
7	6	raleqbi1dv 2693	. . . . . . 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 3344	. . . . . . 7 ⊢ (𝑧 = 𝐷 → (𝐶 ∩ 𝑧) = (𝐶 ∩ 𝐷))
10		sseq2 3193	. . . . . . . . . 10 ⊢ ((𝐶 ∩ 𝑧) = (𝐶 ∩ 𝐷) → (𝑥 ⊆ (𝐶 ∩ 𝑧) ↔ 𝑥 ⊆ (𝐶 ∩ 𝐷)))
11	10	anbi2d 464	. . . . . . . . 9 ⊢ ((𝐶 ∩ 𝑧) = (𝐶 ∩ 𝐷) → ((𝑤 ∈ 𝑥 ∧ 𝑥 ⊆ (𝐶 ∩ 𝑧)) ↔ (𝑤 ∈ 𝑥 ∧ 𝑥 ⊆ (𝐶 ∩ 𝐷))))
12	11	rexbidv 2490	. . . . . . . 8 ⊢ ((𝐶 ∩ 𝑧) = (𝐶 ∩ 𝐷) → (∃𝑥 ∈ 𝐵 (𝑤 ∈ 𝑥 ∧ 𝑥 ⊆ (𝐶 ∩ 𝑧)) ↔ ∃𝑥 ∈ 𝐵 (𝑤 ∈ 𝑥 ∧ 𝑥 ⊆ (𝐶 ∩ 𝐷))))
13	12	raleqbi1dv 2693	. . . . . . 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 2868	. . . . 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 2251	. . . . . . . 8 ⊢ (𝑤 = 𝐴 → (𝑤 ∈ 𝑥 ↔ 𝐴 ∈ 𝑥))
17	16	anbi1d 465	. . . . . . 7 ⊢ (𝑤 = 𝐴 → ((𝑤 ∈ 𝑥 ∧ 𝑥 ⊆ (𝐶 ∩ 𝐷)) ↔ (𝐴 ∈ 𝑥 ∧ 𝑥 ⊆ (𝐶 ∩ 𝐷))))
18	17	rexbidv 2490	. . . . . 6 ⊢ (𝑤 = 𝐴 → (∃𝑥 ∈ 𝐵 (𝑤 ∈ 𝑥 ∧ 𝑥 ⊆ (𝐶 ∩ 𝐷)) ↔ ∃𝑥 ∈ 𝐵 (𝐴 ∈ 𝑥 ∧ 𝑥 ⊆ (𝐶 ∩ 𝐷))))
19	18	rspccv 2852	. . . . 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 258	. 2 ⊢ (𝐵 ∈ TopBases → (𝐶 ∈ 𝐵 → (𝐷 ∈ 𝐵 → (𝐴 ∈ (𝐶 ∩ 𝐷) → ∃𝑥 ∈ 𝐵 (𝐴 ∈ 𝑥 ∧ 𝑥 ⊆ (𝐶 ∩ 𝐷))))))
23	22	imp43 355	1 ⊢ (((𝐵 ∈ TopBases ∧ 𝐶 ∈ 𝐵) ∧ (𝐷 ∈ 𝐵 ∧ 𝐴 ∈ (𝐶 ∩ 𝐷))) → ∃𝑥 ∈ 𝐵 (𝐴 ∈ 𝑥 ∧ 𝑥 ⊆ (𝐶 ∩ 𝐷)))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1363   ∈ wcel 2159  ∀wral 2467  ∃wrex 2468   ∩ cin 3142   ⊆ wss 3143  TopBasesctb 13925

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 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2170

This theorem depends on definitions:  df-bi 117  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2175  df-cleq 2181  df-clel 2184  df-nfc 2320  df-ral 2472  df-rex 2473  df-v 2753  df-in 3149  df-ss 3156  df-pw 3591  df-uni 3824  df-bases 13926

This theorem is referenced by:  tgcl  13947  restbasg  14051  txbas  14141  tgioo  14429