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Theorem basis2 20846
Description: Property of a basis. (Contributed by NM, 17-Jul-2006.)
Assertion
Ref Expression
basis2 (((𝐵 ∈ TopBases ∧ 𝐶𝐵) ∧ (𝐷𝐵𝐴 ∈ (𝐶𝐷))) → ∃𝑥𝐵 (𝐴𝑥𝑥 ⊆ (𝐶𝐷)))
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.
StepHypRef Expression
1 isbasis2g 20843 . . . . 5 (𝐵 ∈ TopBases → (𝐵 ∈ TopBases ↔ ∀𝑦𝐵𝑧𝐵𝑤 ∈ (𝑦𝑧)∃𝑥𝐵 (𝑤𝑥𝑥 ⊆ (𝑦𝑧))))
21ibi 256 . . . 4 (𝐵 ∈ TopBases → ∀𝑦𝐵𝑧𝐵𝑤 ∈ (𝑦𝑧)∃𝑥𝐵 (𝑤𝑥𝑥 ⊆ (𝑦𝑧)))
3 ineq1 3883 . . . . . . 7 (𝑦 = 𝐶 → (𝑦𝑧) = (𝐶𝑧))
4 sseq2 3701 . . . . . . . . . 10 ((𝑦𝑧) = (𝐶𝑧) → (𝑥 ⊆ (𝑦𝑧) ↔ 𝑥 ⊆ (𝐶𝑧)))
54anbi2d 742 . . . . . . . . 9 ((𝑦𝑧) = (𝐶𝑧) → ((𝑤𝑥𝑥 ⊆ (𝑦𝑧)) ↔ (𝑤𝑥𝑥 ⊆ (𝐶𝑧))))
65rexbidv 3122 . . . . . . . 8 ((𝑦𝑧) = (𝐶𝑧) → (∃𝑥𝐵 (𝑤𝑥𝑥 ⊆ (𝑦𝑧)) ↔ ∃𝑥𝐵 (𝑤𝑥𝑥 ⊆ (𝐶𝑧))))
76raleqbi1dv 3217 . . . . . . 7 ((𝑦𝑧) = (𝐶𝑧) → (∀𝑤 ∈ (𝑦𝑧)∃𝑥𝐵 (𝑤𝑥𝑥 ⊆ (𝑦𝑧)) ↔ ∀𝑤 ∈ (𝐶𝑧)∃𝑥𝐵 (𝑤𝑥𝑥 ⊆ (𝐶𝑧))))
83, 7syl 17 . . . . . 6 (𝑦 = 𝐶 → (∀𝑤 ∈ (𝑦𝑧)∃𝑥𝐵 (𝑤𝑥𝑥 ⊆ (𝑦𝑧)) ↔ ∀𝑤 ∈ (𝐶𝑧)∃𝑥𝐵 (𝑤𝑥𝑥 ⊆ (𝐶𝑧))))
9 ineq2 3884 . . . . . . 7 (𝑧 = 𝐷 → (𝐶𝑧) = (𝐶𝐷))
10 sseq2 3701 . . . . . . . . . 10 ((𝐶𝑧) = (𝐶𝐷) → (𝑥 ⊆ (𝐶𝑧) ↔ 𝑥 ⊆ (𝐶𝐷)))
1110anbi2d 742 . . . . . . . . 9 ((𝐶𝑧) = (𝐶𝐷) → ((𝑤𝑥𝑥 ⊆ (𝐶𝑧)) ↔ (𝑤𝑥𝑥 ⊆ (𝐶𝐷))))
1211rexbidv 3122 . . . . . . . 8 ((𝐶𝑧) = (𝐶𝐷) → (∃𝑥𝐵 (𝑤𝑥𝑥 ⊆ (𝐶𝑧)) ↔ ∃𝑥𝐵 (𝑤𝑥𝑥 ⊆ (𝐶𝐷))))
1312raleqbi1dv 3217 . . . . . . 7 ((𝐶𝑧) = (𝐶𝐷) → (∀𝑤 ∈ (𝐶𝑧)∃𝑥𝐵 (𝑤𝑥𝑥 ⊆ (𝐶𝑧)) ↔ ∀𝑤 ∈ (𝐶𝐷)∃𝑥𝐵 (𝑤𝑥𝑥 ⊆ (𝐶𝐷))))
149, 13syl 17 . . . . . 6 (𝑧 = 𝐷 → (∀𝑤 ∈ (𝐶𝑧)∃𝑥𝐵 (𝑤𝑥𝑥 ⊆ (𝐶𝑧)) ↔ ∀𝑤 ∈ (𝐶𝐷)∃𝑥𝐵 (𝑤𝑥𝑥 ⊆ (𝐶𝐷))))
158, 14rspc2v 3394 . . . . 5 ((𝐶𝐵𝐷𝐵) → (∀𝑦𝐵𝑧𝐵𝑤 ∈ (𝑦𝑧)∃𝑥𝐵 (𝑤𝑥𝑥 ⊆ (𝑦𝑧)) → ∀𝑤 ∈ (𝐶𝐷)∃𝑥𝐵 (𝑤𝑥𝑥 ⊆ (𝐶𝐷))))
16 eleq1 2759 . . . . . . . 8 (𝑤 = 𝐴 → (𝑤𝑥𝐴𝑥))
1716anbi1d 743 . . . . . . 7 (𝑤 = 𝐴 → ((𝑤𝑥𝑥 ⊆ (𝐶𝐷)) ↔ (𝐴𝑥𝑥 ⊆ (𝐶𝐷))))
1817rexbidv 3122 . . . . . 6 (𝑤 = 𝐴 → (∃𝑥𝐵 (𝑤𝑥𝑥 ⊆ (𝐶𝐷)) ↔ ∃𝑥𝐵 (𝐴𝑥𝑥 ⊆ (𝐶𝐷))))
1918rspccv 3378 . . . . 5 (∀𝑤 ∈ (𝐶𝐷)∃𝑥𝐵 (𝑤𝑥𝑥 ⊆ (𝐶𝐷)) → (𝐴 ∈ (𝐶𝐷) → ∃𝑥𝐵 (𝐴𝑥𝑥 ⊆ (𝐶𝐷))))
2015, 19syl6com 37 . . . 4 (∀𝑦𝐵𝑧𝐵𝑤 ∈ (𝑦𝑧)∃𝑥𝐵 (𝑤𝑥𝑥 ⊆ (𝑦𝑧)) → ((𝐶𝐵𝐷𝐵) → (𝐴 ∈ (𝐶𝐷) → ∃𝑥𝐵 (𝐴𝑥𝑥 ⊆ (𝐶𝐷)))))
212, 20syl 17 . . 3 (𝐵 ∈ TopBases → ((𝐶𝐵𝐷𝐵) → (𝐴 ∈ (𝐶𝐷) → ∃𝑥𝐵 (𝐴𝑥𝑥 ⊆ (𝐶𝐷)))))
2221expd 451 . 2 (𝐵 ∈ TopBases → (𝐶𝐵 → (𝐷𝐵 → (𝐴 ∈ (𝐶𝐷) → ∃𝑥𝐵 (𝐴𝑥𝑥 ⊆ (𝐶𝐷))))))
2322imp43 622 1 (((𝐵 ∈ TopBases ∧ 𝐶𝐵) ∧ (𝐷𝐵𝐴 ∈ (𝐶𝐷))) → ∃𝑥𝐵 (𝐴𝑥𝑥 ⊆ (𝐶𝐷)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1564  wcel 2071  wral 2982  wrex 2983  cin 3647  wss 3648  TopBasesctb 20840
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1818  ax-5 1920  ax-6 1986  ax-7 2022  ax-9 2080  ax-10 2100  ax-11 2115  ax-12 2128  ax-13 2323  ax-ext 2672
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1567  df-ex 1786  df-nf 1791  df-sb 1979  df-clab 2679  df-cleq 2685  df-clel 2688  df-nfc 2823  df-ral 2987  df-rex 2988  df-v 3274  df-in 3655  df-ss 3662  df-pw 4236  df-uni 4513  df-bases 20841
This theorem is referenced by:  tgcl  20864  restbas  21053  txbas  21461  basqtop  21605  tgioo  22689
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