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Mirrors > Home > MPE Home > Th. List > fbasssin | Structured version Visualization version GIF version |
Description: A filter base contains subsets of its pairwise intersections. (Contributed by Jeff Hankins, 1-Sep-2009.) (Revised by Jeff Hankins, 1-Dec-2010.) |
Ref | Expression |
---|---|
fbasssin | ⊢ ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐴 ∈ 𝐹 ∧ 𝐵 ∈ 𝐹) → ∃𝑥 ∈ 𝐹 𝑥 ⊆ (𝐴 ∩ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elfvdm 6928 | . . . . . . 7 ⊢ (𝐹 ∈ (fBas‘𝑋) → 𝑋 ∈ dom fBas) | |
2 | isfbas2 23658 | . . . . . . 7 ⊢ (𝑋 ∈ dom fBas → (𝐹 ∈ (fBas‘𝑋) ↔ (𝐹 ⊆ 𝒫 𝑋 ∧ (𝐹 ≠ ∅ ∧ ∅ ∉ 𝐹 ∧ ∀𝑦 ∈ 𝐹 ∀𝑧 ∈ 𝐹 ∃𝑥 ∈ 𝐹 𝑥 ⊆ (𝑦 ∩ 𝑧))))) | |
3 | 1, 2 | syl 17 | . . . . . 6 ⊢ (𝐹 ∈ (fBas‘𝑋) → (𝐹 ∈ (fBas‘𝑋) ↔ (𝐹 ⊆ 𝒫 𝑋 ∧ (𝐹 ≠ ∅ ∧ ∅ ∉ 𝐹 ∧ ∀𝑦 ∈ 𝐹 ∀𝑧 ∈ 𝐹 ∃𝑥 ∈ 𝐹 𝑥 ⊆ (𝑦 ∩ 𝑧))))) |
4 | 3 | ibi 267 | . . . . 5 ⊢ (𝐹 ∈ (fBas‘𝑋) → (𝐹 ⊆ 𝒫 𝑋 ∧ (𝐹 ≠ ∅ ∧ ∅ ∉ 𝐹 ∧ ∀𝑦 ∈ 𝐹 ∀𝑧 ∈ 𝐹 ∃𝑥 ∈ 𝐹 𝑥 ⊆ (𝑦 ∩ 𝑧)))) |
5 | 4 | simprd 495 | . . . 4 ⊢ (𝐹 ∈ (fBas‘𝑋) → (𝐹 ≠ ∅ ∧ ∅ ∉ 𝐹 ∧ ∀𝑦 ∈ 𝐹 ∀𝑧 ∈ 𝐹 ∃𝑥 ∈ 𝐹 𝑥 ⊆ (𝑦 ∩ 𝑧))) |
6 | 5 | simp3d 1143 | . . 3 ⊢ (𝐹 ∈ (fBas‘𝑋) → ∀𝑦 ∈ 𝐹 ∀𝑧 ∈ 𝐹 ∃𝑥 ∈ 𝐹 𝑥 ⊆ (𝑦 ∩ 𝑧)) |
7 | ineq1 4205 | . . . . . 6 ⊢ (𝑦 = 𝐴 → (𝑦 ∩ 𝑧) = (𝐴 ∩ 𝑧)) | |
8 | 7 | sseq2d 4014 | . . . . 5 ⊢ (𝑦 = 𝐴 → (𝑥 ⊆ (𝑦 ∩ 𝑧) ↔ 𝑥 ⊆ (𝐴 ∩ 𝑧))) |
9 | 8 | rexbidv 3177 | . . . 4 ⊢ (𝑦 = 𝐴 → (∃𝑥 ∈ 𝐹 𝑥 ⊆ (𝑦 ∩ 𝑧) ↔ ∃𝑥 ∈ 𝐹 𝑥 ⊆ (𝐴 ∩ 𝑧))) |
10 | ineq2 4206 | . . . . . 6 ⊢ (𝑧 = 𝐵 → (𝐴 ∩ 𝑧) = (𝐴 ∩ 𝐵)) | |
11 | 10 | sseq2d 4014 | . . . . 5 ⊢ (𝑧 = 𝐵 → (𝑥 ⊆ (𝐴 ∩ 𝑧) ↔ 𝑥 ⊆ (𝐴 ∩ 𝐵))) |
12 | 11 | rexbidv 3177 | . . . 4 ⊢ (𝑧 = 𝐵 → (∃𝑥 ∈ 𝐹 𝑥 ⊆ (𝐴 ∩ 𝑧) ↔ ∃𝑥 ∈ 𝐹 𝑥 ⊆ (𝐴 ∩ 𝐵))) |
13 | 9, 12 | rspc2v 3622 | . . 3 ⊢ ((𝐴 ∈ 𝐹 ∧ 𝐵 ∈ 𝐹) → (∀𝑦 ∈ 𝐹 ∀𝑧 ∈ 𝐹 ∃𝑥 ∈ 𝐹 𝑥 ⊆ (𝑦 ∩ 𝑧) → ∃𝑥 ∈ 𝐹 𝑥 ⊆ (𝐴 ∩ 𝐵))) |
14 | 6, 13 | syl5com 31 | . 2 ⊢ (𝐹 ∈ (fBas‘𝑋) → ((𝐴 ∈ 𝐹 ∧ 𝐵 ∈ 𝐹) → ∃𝑥 ∈ 𝐹 𝑥 ⊆ (𝐴 ∩ 𝐵))) |
15 | 14 | 3impib 1115 | 1 ⊢ ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐴 ∈ 𝐹 ∧ 𝐵 ∈ 𝐹) → ∃𝑥 ∈ 𝐹 𝑥 ⊆ (𝐴 ∩ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 ∧ w3a 1086 = wceq 1540 ∈ wcel 2105 ≠ wne 2939 ∉ wnel 3045 ∀wral 3060 ∃wrex 3069 ∩ cin 3947 ⊆ wss 3948 ∅c0 4322 𝒫 cpw 4602 dom cdm 5676 ‘cfv 6543 fBascfbas 21220 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-iota 6495 df-fun 6545 df-fv 6551 df-fbas 21229 |
This theorem is referenced by: fbssfi 23660 fbncp 23662 fbun 23663 fbfinnfr 23664 trfbas2 23666 filin 23677 fgcl 23701 fbasrn 23707 |
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