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Mirrors > Home > MPE Home > Th. List > fbasweak | Structured version Visualization version GIF version |
Description: A filter base on any set is also a filter base on any larger set. (Contributed by Stefan O'Rear, 2-Aug-2015.) |
Ref | Expression |
---|---|
fbasweak | ⊢ ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐹 ⊆ 𝒫 𝑌 ∧ 𝑌 ∈ 𝑉) → 𝐹 ∈ (fBas‘𝑌)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp2 1128 | . 2 ⊢ ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐹 ⊆ 𝒫 𝑌 ∧ 𝑌 ∈ 𝑉) → 𝐹 ⊆ 𝒫 𝑌) | |
2 | simp1 1127 | . . . 4 ⊢ ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐹 ⊆ 𝒫 𝑌 ∧ 𝑌 ∈ 𝑉) → 𝐹 ∈ (fBas‘𝑋)) | |
3 | elfvdm 6480 | . . . . . 6 ⊢ (𝐹 ∈ (fBas‘𝑋) → 𝑋 ∈ dom fBas) | |
4 | 3 | 3ad2ant1 1124 | . . . . 5 ⊢ ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐹 ⊆ 𝒫 𝑌 ∧ 𝑌 ∈ 𝑉) → 𝑋 ∈ dom fBas) |
5 | isfbas 22052 | . . . . 5 ⊢ (𝑋 ∈ dom fBas → (𝐹 ∈ (fBas‘𝑋) ↔ (𝐹 ⊆ 𝒫 𝑋 ∧ (𝐹 ≠ ∅ ∧ ∅ ∉ 𝐹 ∧ ∀𝑥 ∈ 𝐹 ∀𝑦 ∈ 𝐹 (𝐹 ∩ 𝒫 (𝑥 ∩ 𝑦)) ≠ ∅)))) | |
6 | 4, 5 | syl 17 | . . . 4 ⊢ ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐹 ⊆ 𝒫 𝑌 ∧ 𝑌 ∈ 𝑉) → (𝐹 ∈ (fBas‘𝑋) ↔ (𝐹 ⊆ 𝒫 𝑋 ∧ (𝐹 ≠ ∅ ∧ ∅ ∉ 𝐹 ∧ ∀𝑥 ∈ 𝐹 ∀𝑦 ∈ 𝐹 (𝐹 ∩ 𝒫 (𝑥 ∩ 𝑦)) ≠ ∅)))) |
7 | 2, 6 | mpbid 224 | . . 3 ⊢ ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐹 ⊆ 𝒫 𝑌 ∧ 𝑌 ∈ 𝑉) → (𝐹 ⊆ 𝒫 𝑋 ∧ (𝐹 ≠ ∅ ∧ ∅ ∉ 𝐹 ∧ ∀𝑥 ∈ 𝐹 ∀𝑦 ∈ 𝐹 (𝐹 ∩ 𝒫 (𝑥 ∩ 𝑦)) ≠ ∅))) |
8 | 7 | simprd 491 | . 2 ⊢ ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐹 ⊆ 𝒫 𝑌 ∧ 𝑌 ∈ 𝑉) → (𝐹 ≠ ∅ ∧ ∅ ∉ 𝐹 ∧ ∀𝑥 ∈ 𝐹 ∀𝑦 ∈ 𝐹 (𝐹 ∩ 𝒫 (𝑥 ∩ 𝑦)) ≠ ∅)) |
9 | isfbas 22052 | . . 3 ⊢ (𝑌 ∈ 𝑉 → (𝐹 ∈ (fBas‘𝑌) ↔ (𝐹 ⊆ 𝒫 𝑌 ∧ (𝐹 ≠ ∅ ∧ ∅ ∉ 𝐹 ∧ ∀𝑥 ∈ 𝐹 ∀𝑦 ∈ 𝐹 (𝐹 ∩ 𝒫 (𝑥 ∩ 𝑦)) ≠ ∅)))) | |
10 | 9 | 3ad2ant3 1126 | . 2 ⊢ ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐹 ⊆ 𝒫 𝑌 ∧ 𝑌 ∈ 𝑉) → (𝐹 ∈ (fBas‘𝑌) ↔ (𝐹 ⊆ 𝒫 𝑌 ∧ (𝐹 ≠ ∅ ∧ ∅ ∉ 𝐹 ∧ ∀𝑥 ∈ 𝐹 ∀𝑦 ∈ 𝐹 (𝐹 ∩ 𝒫 (𝑥 ∩ 𝑦)) ≠ ∅)))) |
11 | 1, 8, 10 | mpbir2and 703 | 1 ⊢ ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐹 ⊆ 𝒫 𝑌 ∧ 𝑌 ∈ 𝑉) → 𝐹 ∈ (fBas‘𝑌)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 386 ∧ w3a 1071 ∈ wcel 2107 ≠ wne 2969 ∉ wnel 3075 ∀wral 3090 ∩ cin 3791 ⊆ wss 3792 ∅c0 4141 𝒫 cpw 4379 dom cdm 5357 ‘cfv 6137 fBascfbas 20141 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-sep 5019 ax-nul 5027 ax-pow 5079 ax-pr 5140 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-op 4405 df-uni 4674 df-br 4889 df-opab 4951 df-mpt 4968 df-id 5263 df-xp 5363 df-rel 5364 df-cnv 5365 df-co 5366 df-dm 5367 df-rn 5368 df-res 5369 df-ima 5370 df-iota 6101 df-fun 6139 df-fv 6145 df-fbas 20150 |
This theorem is referenced by: snfbas 22089 fgabs 22102 fgtr 22113 trfg 22114 ssufl 22141 cfiluweak 22518 cfilresi 23512 cmetss 23533 minveclem4a 23647 minveclem4 23649 |
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