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Theorem fbasssin 23844
Description: A filter base contains subsets of its pairwise intersections. (Contributed by Jeff Hankins, 1-Sep-2009.) (Revised by Jeff Hankins, 1-Dec-2010.)
Assertion
Ref Expression
fbasssin ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐴𝐹𝐵𝐹) → ∃𝑥𝐹 𝑥 ⊆ (𝐴𝐵))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝐹   𝑥,𝑋

Proof of Theorem fbasssin
Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elfvdm 6943 . . . . . . 7 (𝐹 ∈ (fBas‘𝑋) → 𝑋 ∈ dom fBas)
2 isfbas2 23843 . . . . . . 7 (𝑋 ∈ dom fBas → (𝐹 ∈ (fBas‘𝑋) ↔ (𝐹 ⊆ 𝒫 𝑋 ∧ (𝐹 ≠ ∅ ∧ ∅ ∉ 𝐹 ∧ ∀𝑦𝐹𝑧𝐹𝑥𝐹 𝑥 ⊆ (𝑦𝑧)))))
31, 2syl 17 . . . . . 6 (𝐹 ∈ (fBas‘𝑋) → (𝐹 ∈ (fBas‘𝑋) ↔ (𝐹 ⊆ 𝒫 𝑋 ∧ (𝐹 ≠ ∅ ∧ ∅ ∉ 𝐹 ∧ ∀𝑦𝐹𝑧𝐹𝑥𝐹 𝑥 ⊆ (𝑦𝑧)))))
43ibi 267 . . . . 5 (𝐹 ∈ (fBas‘𝑋) → (𝐹 ⊆ 𝒫 𝑋 ∧ (𝐹 ≠ ∅ ∧ ∅ ∉ 𝐹 ∧ ∀𝑦𝐹𝑧𝐹𝑥𝐹 𝑥 ⊆ (𝑦𝑧))))
54simprd 495 . . . 4 (𝐹 ∈ (fBas‘𝑋) → (𝐹 ≠ ∅ ∧ ∅ ∉ 𝐹 ∧ ∀𝑦𝐹𝑧𝐹𝑥𝐹 𝑥 ⊆ (𝑦𝑧)))
65simp3d 1145 . . 3 (𝐹 ∈ (fBas‘𝑋) → ∀𝑦𝐹𝑧𝐹𝑥𝐹 𝑥 ⊆ (𝑦𝑧))
7 ineq1 4213 . . . . . 6 (𝑦 = 𝐴 → (𝑦𝑧) = (𝐴𝑧))
87sseq2d 4016 . . . . 5 (𝑦 = 𝐴 → (𝑥 ⊆ (𝑦𝑧) ↔ 𝑥 ⊆ (𝐴𝑧)))
98rexbidv 3179 . . . 4 (𝑦 = 𝐴 → (∃𝑥𝐹 𝑥 ⊆ (𝑦𝑧) ↔ ∃𝑥𝐹 𝑥 ⊆ (𝐴𝑧)))
10 ineq2 4214 . . . . . 6 (𝑧 = 𝐵 → (𝐴𝑧) = (𝐴𝐵))
1110sseq2d 4016 . . . . 5 (𝑧 = 𝐵 → (𝑥 ⊆ (𝐴𝑧) ↔ 𝑥 ⊆ (𝐴𝐵)))
1211rexbidv 3179 . . . 4 (𝑧 = 𝐵 → (∃𝑥𝐹 𝑥 ⊆ (𝐴𝑧) ↔ ∃𝑥𝐹 𝑥 ⊆ (𝐴𝐵)))
139, 12rspc2v 3633 . . 3 ((𝐴𝐹𝐵𝐹) → (∀𝑦𝐹𝑧𝐹𝑥𝐹 𝑥 ⊆ (𝑦𝑧) → ∃𝑥𝐹 𝑥 ⊆ (𝐴𝐵)))
146, 13syl5com 31 . 2 (𝐹 ∈ (fBas‘𝑋) → ((𝐴𝐹𝐵𝐹) → ∃𝑥𝐹 𝑥 ⊆ (𝐴𝐵)))
15143impib 1117 1 ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐴𝐹𝐵𝐹) → ∃𝑥𝐹 𝑥 ⊆ (𝐴𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1087   = wceq 1540  wcel 2108  wne 2940  wnel 3046  wral 3061  wrex 3070  cin 3950  wss 3951  c0 4333  𝒫 cpw 4600  dom cdm 5685  cfv 6561  fBascfbas 21352
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fv 6569  df-fbas 21361
This theorem is referenced by:  fbssfi  23845  fbncp  23847  fbun  23848  fbfinnfr  23849  trfbas2  23851  filin  23862  fgcl  23886  fbasrn  23892
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