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Theorem basis2 12805
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 12802 . . . . 5 (𝐵 ∈ TopBases → (𝐵 ∈ TopBases ↔ ∀𝑦𝐵𝑧𝐵𝑤 ∈ (𝑦𝑧)∃𝑥𝐵 (𝑤𝑥𝑥 ⊆ (𝑦𝑧))))
21ibi 175 . . . 4 (𝐵 ∈ TopBases → ∀𝑦𝐵𝑧𝐵𝑤 ∈ (𝑦𝑧)∃𝑥𝐵 (𝑤𝑥𝑥 ⊆ (𝑦𝑧)))
3 ineq1 3321 . . . . . . 7 (𝑦 = 𝐶 → (𝑦𝑧) = (𝐶𝑧))
4 sseq2 3171 . . . . . . . . . 10 ((𝑦𝑧) = (𝐶𝑧) → (𝑥 ⊆ (𝑦𝑧) ↔ 𝑥 ⊆ (𝐶𝑧)))
54anbi2d 461 . . . . . . . . 9 ((𝑦𝑧) = (𝐶𝑧) → ((𝑤𝑥𝑥 ⊆ (𝑦𝑧)) ↔ (𝑤𝑥𝑥 ⊆ (𝐶𝑧))))
65rexbidv 2471 . . . . . . . 8 ((𝑦𝑧) = (𝐶𝑧) → (∃𝑥𝐵 (𝑤𝑥𝑥 ⊆ (𝑦𝑧)) ↔ ∃𝑥𝐵 (𝑤𝑥𝑥 ⊆ (𝐶𝑧))))
76raleqbi1dv 2673 . . . . . . 7 ((𝑦𝑧) = (𝐶𝑧) → (∀𝑤 ∈ (𝑦𝑧)∃𝑥𝐵 (𝑤𝑥𝑥 ⊆ (𝑦𝑧)) ↔ ∀𝑤 ∈ (𝐶𝑧)∃𝑥𝐵 (𝑤𝑥𝑥 ⊆ (𝐶𝑧))))
83, 7syl 14 . . . . . 6 (𝑦 = 𝐶 → (∀𝑤 ∈ (𝑦𝑧)∃𝑥𝐵 (𝑤𝑥𝑥 ⊆ (𝑦𝑧)) ↔ ∀𝑤 ∈ (𝐶𝑧)∃𝑥𝐵 (𝑤𝑥𝑥 ⊆ (𝐶𝑧))))
9 ineq2 3322 . . . . . . 7 (𝑧 = 𝐷 → (𝐶𝑧) = (𝐶𝐷))
10 sseq2 3171 . . . . . . . . . 10 ((𝐶𝑧) = (𝐶𝐷) → (𝑥 ⊆ (𝐶𝑧) ↔ 𝑥 ⊆ (𝐶𝐷)))
1110anbi2d 461 . . . . . . . . 9 ((𝐶𝑧) = (𝐶𝐷) → ((𝑤𝑥𝑥 ⊆ (𝐶𝑧)) ↔ (𝑤𝑥𝑥 ⊆ (𝐶𝐷))))
1211rexbidv 2471 . . . . . . . 8 ((𝐶𝑧) = (𝐶𝐷) → (∃𝑥𝐵 (𝑤𝑥𝑥 ⊆ (𝐶𝑧)) ↔ ∃𝑥𝐵 (𝑤𝑥𝑥 ⊆ (𝐶𝐷))))
1312raleqbi1dv 2673 . . . . . . 7 ((𝐶𝑧) = (𝐶𝐷) → (∀𝑤 ∈ (𝐶𝑧)∃𝑥𝐵 (𝑤𝑥𝑥 ⊆ (𝐶𝑧)) ↔ ∀𝑤 ∈ (𝐶𝐷)∃𝑥𝐵 (𝑤𝑥𝑥 ⊆ (𝐶𝐷))))
149, 13syl 14 . . . . . 6 (𝑧 = 𝐷 → (∀𝑤 ∈ (𝐶𝑧)∃𝑥𝐵 (𝑤𝑥𝑥 ⊆ (𝐶𝑧)) ↔ ∀𝑤 ∈ (𝐶𝐷)∃𝑥𝐵 (𝑤𝑥𝑥 ⊆ (𝐶𝐷))))
158, 14rspc2v 2847 . . . . 5 ((𝐶𝐵𝐷𝐵) → (∀𝑦𝐵𝑧𝐵𝑤 ∈ (𝑦𝑧)∃𝑥𝐵 (𝑤𝑥𝑥 ⊆ (𝑦𝑧)) → ∀𝑤 ∈ (𝐶𝐷)∃𝑥𝐵 (𝑤𝑥𝑥 ⊆ (𝐶𝐷))))
16 eleq1 2233 . . . . . . . 8 (𝑤 = 𝐴 → (𝑤𝑥𝐴𝑥))
1716anbi1d 462 . . . . . . 7 (𝑤 = 𝐴 → ((𝑤𝑥𝑥 ⊆ (𝐶𝐷)) ↔ (𝐴𝑥𝑥 ⊆ (𝐶𝐷))))
1817rexbidv 2471 . . . . . 6 (𝑤 = 𝐴 → (∃𝑥𝐵 (𝑤𝑥𝑥 ⊆ (𝐶𝐷)) ↔ ∃𝑥𝐵 (𝐴𝑥𝑥 ⊆ (𝐶𝐷))))
1918rspccv 2831 . . . . 5 (∀𝑤 ∈ (𝐶𝐷)∃𝑥𝐵 (𝑤𝑥𝑥 ⊆ (𝐶𝐷)) → (𝐴 ∈ (𝐶𝐷) → ∃𝑥𝐵 (𝐴𝑥𝑥 ⊆ (𝐶𝐷))))
2015, 19syl6com 35 . . . 4 (∀𝑦𝐵𝑧𝐵𝑤 ∈ (𝑦𝑧)∃𝑥𝐵 (𝑤𝑥𝑥 ⊆ (𝑦𝑧)) → ((𝐶𝐵𝐷𝐵) → (𝐴 ∈ (𝐶𝐷) → ∃𝑥𝐵 (𝐴𝑥𝑥 ⊆ (𝐶𝐷)))))
212, 20syl 14 . . 3 (𝐵 ∈ TopBases → ((𝐶𝐵𝐷𝐵) → (𝐴 ∈ (𝐶𝐷) → ∃𝑥𝐵 (𝐴𝑥𝑥 ⊆ (𝐶𝐷)))))
2221expd 256 . 2 (𝐵 ∈ TopBases → (𝐶𝐵 → (𝐷𝐵 → (𝐴 ∈ (𝐶𝐷) → ∃𝑥𝐵 (𝐴𝑥𝑥 ⊆ (𝐶𝐷))))))
2322imp43 353 1 (((𝐵 ∈ TopBases ∧ 𝐶𝐵) ∧ (𝐷𝐵𝐴 ∈ (𝐶𝐷))) → ∃𝑥𝐵 (𝐴𝑥𝑥 ⊆ (𝐶𝐷)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104   = wceq 1348  wcel 2141  wral 2448  wrex 2449  cin 3120  wss 3121  TopBasesctb 12799
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-in 3127  df-ss 3134  df-pw 3566  df-uni 3795  df-bases 12800
This theorem is referenced by:  tgcl  12823  restbasg  12927  txbas  13017  tgioo  13305
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