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Theorem fbasssin 22443
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 6701 . . . . . . 7 (𝐹 ∈ (fBas‘𝑋) → 𝑋 ∈ dom fBas)
2 isfbas2 22442 . . . . . . 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 269 . . . . 5 (𝐹 ∈ (fBas‘𝑋) → (𝐹 ⊆ 𝒫 𝑋 ∧ (𝐹 ≠ ∅ ∧ ∅ ∉ 𝐹 ∧ ∀𝑦𝐹𝑧𝐹𝑥𝐹 𝑥 ⊆ (𝑦𝑧))))
54simprd 498 . . . 4 (𝐹 ∈ (fBas‘𝑋) → (𝐹 ≠ ∅ ∧ ∅ ∉ 𝐹 ∧ ∀𝑦𝐹𝑧𝐹𝑥𝐹 𝑥 ⊆ (𝑦𝑧)))
65simp3d 1140 . . 3 (𝐹 ∈ (fBas‘𝑋) → ∀𝑦𝐹𝑧𝐹𝑥𝐹 𝑥 ⊆ (𝑦𝑧))
7 ineq1 4180 . . . . . 6 (𝑦 = 𝐴 → (𝑦𝑧) = (𝐴𝑧))
87sseq2d 3998 . . . . 5 (𝑦 = 𝐴 → (𝑥 ⊆ (𝑦𝑧) ↔ 𝑥 ⊆ (𝐴𝑧)))
98rexbidv 3297 . . . 4 (𝑦 = 𝐴 → (∃𝑥𝐹 𝑥 ⊆ (𝑦𝑧) ↔ ∃𝑥𝐹 𝑥 ⊆ (𝐴𝑧)))
10 ineq2 4182 . . . . . 6 (𝑧 = 𝐵 → (𝐴𝑧) = (𝐴𝐵))
1110sseq2d 3998 . . . . 5 (𝑧 = 𝐵 → (𝑥 ⊆ (𝐴𝑧) ↔ 𝑥 ⊆ (𝐴𝐵)))
1211rexbidv 3297 . . . 4 (𝑧 = 𝐵 → (∃𝑥𝐹 𝑥 ⊆ (𝐴𝑧) ↔ ∃𝑥𝐹 𝑥 ⊆ (𝐴𝐵)))
139, 12rspc2v 3632 . . 3 ((𝐴𝐹𝐵𝐹) → (∀𝑦𝐹𝑧𝐹𝑥𝐹 𝑥 ⊆ (𝑦𝑧) → ∃𝑥𝐹 𝑥 ⊆ (𝐴𝐵)))
146, 13syl5com 31 . 2 (𝐹 ∈ (fBas‘𝑋) → ((𝐴𝐹𝐵𝐹) → ∃𝑥𝐹 𝑥 ⊆ (𝐴𝐵)))
15143impib 1112 1 ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐴𝐹𝐵𝐹) → ∃𝑥𝐹 𝑥 ⊆ (𝐴𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208  wa 398  w3a 1083   = wceq 1533  wcel 2110  wne 3016  wnel 3123  wral 3138  wrex 3139  cin 3934  wss 3935  c0 4290  𝒫 cpw 4538  dom cdm 5554  cfv 6354  fBascfbas 20532
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5202  ax-nul 5209  ax-pow 5265  ax-pr 5329
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-nel 3124  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4567  df-pr 4569  df-op 4573  df-uni 4838  df-br 5066  df-opab 5128  df-mpt 5146  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-iota 6313  df-fun 6356  df-fv 6362  df-fbas 20541
This theorem is referenced by:  fbssfi  22444  fbncp  22446  fbun  22447  fbfinnfr  22448  trfbas2  22450  filin  22461  fgcl  22485  fbasrn  22491
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