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Theorem fbasssin 23203
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 6884 . . . . . . 7 (𝐹 ∈ (fBas‘𝑋) → 𝑋 ∈ dom fBas)
2 isfbas2 23202 . . . . . . 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 497 . . . 4 (𝐹 ∈ (fBas‘𝑋) → (𝐹 ≠ ∅ ∧ ∅ ∉ 𝐹 ∧ ∀𝑦𝐹𝑧𝐹𝑥𝐹 𝑥 ⊆ (𝑦𝑧)))
65simp3d 1145 . . 3 (𝐹 ∈ (fBas‘𝑋) → ∀𝑦𝐹𝑧𝐹𝑥𝐹 𝑥 ⊆ (𝑦𝑧))
7 ineq1 4170 . . . . . 6 (𝑦 = 𝐴 → (𝑦𝑧) = (𝐴𝑧))
87sseq2d 3981 . . . . 5 (𝑦 = 𝐴 → (𝑥 ⊆ (𝑦𝑧) ↔ 𝑥 ⊆ (𝐴𝑧)))
98rexbidv 3176 . . . 4 (𝑦 = 𝐴 → (∃𝑥𝐹 𝑥 ⊆ (𝑦𝑧) ↔ ∃𝑥𝐹 𝑥 ⊆ (𝐴𝑧)))
10 ineq2 4171 . . . . . 6 (𝑧 = 𝐵 → (𝐴𝑧) = (𝐴𝐵))
1110sseq2d 3981 . . . . 5 (𝑧 = 𝐵 → (𝑥 ⊆ (𝐴𝑧) ↔ 𝑥 ⊆ (𝐴𝐵)))
1211rexbidv 3176 . . . 4 (𝑧 = 𝐵 → (∃𝑥𝐹 𝑥 ⊆ (𝐴𝑧) ↔ ∃𝑥𝐹 𝑥 ⊆ (𝐴𝐵)))
139, 12rspc2v 3593 . . 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 205  wa 397  w3a 1088   = wceq 1542  wcel 2107  wne 2944  wnel 3050  wral 3065  wrex 3074  cin 3914  wss 3915  c0 4287  𝒫 cpw 4565  dom cdm 5638  cfv 6501  fBascfbas 20800
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-nel 3051  df-ral 3066  df-rex 3075  df-rab 3411  df-v 3450  df-sbc 3745  df-csb 3861  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-br 5111  df-opab 5173  df-mpt 5194  df-id 5536  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6453  df-fun 6503  df-fv 6509  df-fbas 20809
This theorem is referenced by:  fbssfi  23204  fbncp  23206  fbun  23207  fbfinnfr  23208  trfbas2  23210  filin  23221  fgcl  23245  fbasrn  23251
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