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Mirrors > Home > MPE Home > Th. List > fgval | Structured version Visualization version GIF version |
Description: The filter generating class gives a filter for every filter base. (Contributed by Jeff Hankins, 3-Sep-2009.) (Revised by Stefan O'Rear, 2-Aug-2015.) |
Ref | Expression |
---|---|
fgval | ⊢ (𝐹 ∈ (fBas‘𝑋) → (𝑋filGen𝐹) = {𝑥 ∈ 𝒫 𝑋 ∣ (𝐹 ∩ 𝒫 𝑥) ≠ ∅}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-fg 20471 | . . 3 ⊢ filGen = (𝑣 ∈ V, 𝑓 ∈ (fBas‘𝑣) ↦ {𝑥 ∈ 𝒫 𝑣 ∣ (𝑓 ∩ 𝒫 𝑥) ≠ ∅}) | |
2 | 1 | a1i 11 | . 2 ⊢ (𝐹 ∈ (fBas‘𝑋) → filGen = (𝑣 ∈ V, 𝑓 ∈ (fBas‘𝑣) ↦ {𝑥 ∈ 𝒫 𝑣 ∣ (𝑓 ∩ 𝒫 𝑥) ≠ ∅})) |
3 | pweq 4538 | . . . . 5 ⊢ (𝑣 = 𝑋 → 𝒫 𝑣 = 𝒫 𝑋) | |
4 | 3 | adantr 481 | . . . 4 ⊢ ((𝑣 = 𝑋 ∧ 𝑓 = 𝐹) → 𝒫 𝑣 = 𝒫 𝑋) |
5 | ineq1 4178 | . . . . . 6 ⊢ (𝑓 = 𝐹 → (𝑓 ∩ 𝒫 𝑥) = (𝐹 ∩ 𝒫 𝑥)) | |
6 | 5 | neeq1d 3072 | . . . . 5 ⊢ (𝑓 = 𝐹 → ((𝑓 ∩ 𝒫 𝑥) ≠ ∅ ↔ (𝐹 ∩ 𝒫 𝑥) ≠ ∅)) |
7 | 6 | adantl 482 | . . . 4 ⊢ ((𝑣 = 𝑋 ∧ 𝑓 = 𝐹) → ((𝑓 ∩ 𝒫 𝑥) ≠ ∅ ↔ (𝐹 ∩ 𝒫 𝑥) ≠ ∅)) |
8 | 4, 7 | rabeqbidv 3483 | . . 3 ⊢ ((𝑣 = 𝑋 ∧ 𝑓 = 𝐹) → {𝑥 ∈ 𝒫 𝑣 ∣ (𝑓 ∩ 𝒫 𝑥) ≠ ∅} = {𝑥 ∈ 𝒫 𝑋 ∣ (𝐹 ∩ 𝒫 𝑥) ≠ ∅}) |
9 | 8 | adantl 482 | . 2 ⊢ ((𝐹 ∈ (fBas‘𝑋) ∧ (𝑣 = 𝑋 ∧ 𝑓 = 𝐹)) → {𝑥 ∈ 𝒫 𝑣 ∣ (𝑓 ∩ 𝒫 𝑥) ≠ ∅} = {𝑥 ∈ 𝒫 𝑋 ∣ (𝐹 ∩ 𝒫 𝑥) ≠ ∅}) |
10 | fveq2 6663 | . . 3 ⊢ (𝑣 = 𝑋 → (fBas‘𝑣) = (fBas‘𝑋)) | |
11 | 10 | adantl 482 | . 2 ⊢ ((𝐹 ∈ (fBas‘𝑋) ∧ 𝑣 = 𝑋) → (fBas‘𝑣) = (fBas‘𝑋)) |
12 | elfvex 6696 | . 2 ⊢ (𝐹 ∈ (fBas‘𝑋) → 𝑋 ∈ V) | |
13 | id 22 | . 2 ⊢ (𝐹 ∈ (fBas‘𝑋) → 𝐹 ∈ (fBas‘𝑋)) | |
14 | elfvdm 6695 | . . 3 ⊢ (𝐹 ∈ (fBas‘𝑋) → 𝑋 ∈ dom fBas) | |
15 | pwexg 5270 | . . 3 ⊢ (𝑋 ∈ dom fBas → 𝒫 𝑋 ∈ V) | |
16 | rabexg 5225 | . . 3 ⊢ (𝒫 𝑋 ∈ V → {𝑥 ∈ 𝒫 𝑋 ∣ (𝐹 ∩ 𝒫 𝑥) ≠ ∅} ∈ V) | |
17 | 14, 15, 16 | 3syl 18 | . 2 ⊢ (𝐹 ∈ (fBas‘𝑋) → {𝑥 ∈ 𝒫 𝑋 ∣ (𝐹 ∩ 𝒫 𝑥) ≠ ∅} ∈ V) |
18 | 2, 9, 11, 12, 13, 17 | ovmpodx 7290 | 1 ⊢ (𝐹 ∈ (fBas‘𝑋) → (𝑋filGen𝐹) = {𝑥 ∈ 𝒫 𝑋 ∣ (𝐹 ∩ 𝒫 𝑥) ≠ ∅}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 = wceq 1528 ∈ wcel 2105 ≠ wne 3013 {crab 3139 Vcvv 3492 ∩ cin 3932 ∅c0 4288 𝒫 cpw 4535 dom cdm 5548 ‘cfv 6348 (class class class)co 7145 ∈ cmpo 7147 fBascfbas 20461 filGencfg 20462 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-iota 6307 df-fun 6350 df-fv 6356 df-ov 7148 df-oprab 7149 df-mpo 7150 df-fg 20471 |
This theorem is referenced by: elfg 22407 restmetu 23107 neifg 33616 |
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