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Theorem fbasssin 23860
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 6944 . . . . . . 7 (𝐹 ∈ (fBas‘𝑋) → 𝑋 ∈ dom fBas)
2 isfbas2 23859 . . . . . . 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 1143 . . 3 (𝐹 ∈ (fBas‘𝑋) → ∀𝑦𝐹𝑧𝐹𝑥𝐹 𝑥 ⊆ (𝑦𝑧))
7 ineq1 4221 . . . . . 6 (𝑦 = 𝐴 → (𝑦𝑧) = (𝐴𝑧))
87sseq2d 4028 . . . . 5 (𝑦 = 𝐴 → (𝑥 ⊆ (𝑦𝑧) ↔ 𝑥 ⊆ (𝐴𝑧)))
98rexbidv 3177 . . . 4 (𝑦 = 𝐴 → (∃𝑥𝐹 𝑥 ⊆ (𝑦𝑧) ↔ ∃𝑥𝐹 𝑥 ⊆ (𝐴𝑧)))
10 ineq2 4222 . . . . . 6 (𝑧 = 𝐵 → (𝐴𝑧) = (𝐴𝐵))
1110sseq2d 4028 . . . . 5 (𝑧 = 𝐵 → (𝑥 ⊆ (𝐴𝑧) ↔ 𝑥 ⊆ (𝐴𝐵)))
1211rexbidv 3177 . . . 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 1115 1 ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐴𝐹𝐵𝐹) → ∃𝑥𝐹 𝑥 ⊆ (𝐴𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1086   = wceq 1537  wcel 2106  wne 2938  wnel 3044  wral 3059  wrex 3068  cin 3962  wss 3963  c0 4339  𝒫 cpw 4605  dom cdm 5689  cfv 6563  fBascfbas 21370
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 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fv 6571  df-fbas 21379
This theorem is referenced by:  fbssfi  23861  fbncp  23863  fbun  23864  fbfinnfr  23865  trfbas2  23867  filin  23878  fgcl  23902  fbasrn  23908
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