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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 19565 | . . 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 4111 | . . . . 5 ⊢ (𝑣 = 𝑋 → 𝒫 𝑣 = 𝒫 𝑋) | |
4 | 3 | adantr 480 | . . . 4 ⊢ ((𝑣 = 𝑋 ∧ 𝑓 = 𝐹) → 𝒫 𝑣 = 𝒫 𝑋) |
5 | ineq1 3769 | . . . . . 6 ⊢ (𝑓 = 𝐹 → (𝑓 ∩ 𝒫 𝑥) = (𝐹 ∩ 𝒫 𝑥)) | |
6 | 5 | neeq1d 2841 | . . . . 5 ⊢ (𝑓 = 𝐹 → ((𝑓 ∩ 𝒫 𝑥) ≠ ∅ ↔ (𝐹 ∩ 𝒫 𝑥) ≠ ∅)) |
7 | 6 | adantl 481 | . . . 4 ⊢ ((𝑣 = 𝑋 ∧ 𝑓 = 𝐹) → ((𝑓 ∩ 𝒫 𝑥) ≠ ∅ ↔ (𝐹 ∩ 𝒫 𝑥) ≠ ∅)) |
8 | 4, 7 | rabeqbidv 3168 | . . 3 ⊢ ((𝑣 = 𝑋 ∧ 𝑓 = 𝐹) → {𝑥 ∈ 𝒫 𝑣 ∣ (𝑓 ∩ 𝒫 𝑥) ≠ ∅} = {𝑥 ∈ 𝒫 𝑋 ∣ (𝐹 ∩ 𝒫 𝑥) ≠ ∅}) |
9 | 8 | adantl 481 | . 2 ⊢ ((𝐹 ∈ (fBas‘𝑋) ∧ (𝑣 = 𝑋 ∧ 𝑓 = 𝐹)) → {𝑥 ∈ 𝒫 𝑣 ∣ (𝑓 ∩ 𝒫 𝑥) ≠ ∅} = {𝑥 ∈ 𝒫 𝑋 ∣ (𝐹 ∩ 𝒫 𝑥) ≠ ∅}) |
10 | fveq2 6103 | . . 3 ⊢ (𝑣 = 𝑋 → (fBas‘𝑣) = (fBas‘𝑋)) | |
11 | 10 | adantl 481 | . 2 ⊢ ((𝐹 ∈ (fBas‘𝑋) ∧ 𝑣 = 𝑋) → (fBas‘𝑣) = (fBas‘𝑋)) |
12 | elfvex 6131 | . 2 ⊢ (𝐹 ∈ (fBas‘𝑋) → 𝑋 ∈ V) | |
13 | id 22 | . 2 ⊢ (𝐹 ∈ (fBas‘𝑋) → 𝐹 ∈ (fBas‘𝑋)) | |
14 | elfvdm 6130 | . . 3 ⊢ (𝐹 ∈ (fBas‘𝑋) → 𝑋 ∈ dom fBas) | |
15 | pwexg 4776 | . . 3 ⊢ (𝑋 ∈ dom fBas → 𝒫 𝑋 ∈ V) | |
16 | rabexg 4739 | . . 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 | ovmpt2dx 6685 | 1 ⊢ (𝐹 ∈ (fBas‘𝑋) → (𝑋filGen𝐹) = {𝑥 ∈ 𝒫 𝑋 ∣ (𝐹 ∩ 𝒫 𝑥) ≠ ∅}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 {crab 2900 Vcvv 3173 ∩ cin 3539 ∅c0 3874 𝒫 cpw 4108 dom cdm 5038 ‘cfv 5804 (class class class)co 6549 ↦ cmpt2 6551 fBascfbas 19555 filGencfg 19556 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-iota 5768 df-fun 5806 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-fg 19565 |
This theorem is referenced by: elfg 21485 restmetu 22185 neifg 31536 |
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