Theorem basis2 13633

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 13630	. . . . 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 3331	. . . . . . 7 ⊢ (𝑦 = 𝐶 → (𝑦 ∩ 𝑧) = (𝐶 ∩ 𝑧))
4		sseq2 3181	. . . . . . . . . 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 2478	. . . . . . . 8 ⊢ ((𝑦 ∩ 𝑧) = (𝐶 ∩ 𝑧) → (∃𝑥 ∈ 𝐵 (𝑤 ∈ 𝑥 ∧ 𝑥 ⊆ (𝑦 ∩ 𝑧)) ↔ ∃𝑥 ∈ 𝐵 (𝑤 ∈ 𝑥 ∧ 𝑥 ⊆ (𝐶 ∩ 𝑧))))
7	6	raleqbi1dv 2681	. . . . . . 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 3332	. . . . . . 7 ⊢ (𝑧 = 𝐷 → (𝐶 ∩ 𝑧) = (𝐶 ∩ 𝐷))
10		sseq2 3181	. . . . . . . . . 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 2478	. . . . . . . 8 ⊢ ((𝐶 ∩ 𝑧) = (𝐶 ∩ 𝐷) → (∃𝑥 ∈ 𝐵 (𝑤 ∈ 𝑥 ∧ 𝑥 ⊆ (𝐶 ∩ 𝑧)) ↔ ∃𝑥 ∈ 𝐵 (𝑤 ∈ 𝑥 ∧ 𝑥 ⊆ (𝐶 ∩ 𝐷))))
13	12	raleqbi1dv 2681	. . . . . . 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 2856	. . . . 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 2240	. . . . . . . 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 2478	. . . . . 6 ⊢ (𝑤 = 𝐴 → (∃𝑥 ∈ 𝐵 (𝑤 ∈ 𝑥 ∧ 𝑥 ⊆ (𝐶 ∩ 𝐷)) ↔ ∃𝑥 ∈ 𝐵 (𝐴 ∈ 𝑥 ∧ 𝑥 ⊆ (𝐶 ∩ 𝐷))))
19	18	rspccv 2840	. . . . 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 1353   ∈ wcel 2148  ∀wral 2455  ∃wrex 2456   ∩ cin 3130   ⊆ wss 3131  TopBasesctb 13627

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-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2741  df-in 3137  df-ss 3144  df-pw 3579  df-uni 3812  df-bases 13628

This theorem is referenced by:  tgcl  13649  restbasg  13753  txbas  13843  tgioo  14131