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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 19744 | . . 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 4161 | . . . . 5 ⊢ (𝑣 = 𝑋 → 𝒫 𝑣 = 𝒫 𝑋) | |
4 | 3 | adantr 481 | . . . 4 ⊢ ((𝑣 = 𝑋 ∧ 𝑓 = 𝐹) → 𝒫 𝑣 = 𝒫 𝑋) |
5 | ineq1 3807 | . . . . . 6 ⊢ (𝑓 = 𝐹 → (𝑓 ∩ 𝒫 𝑥) = (𝐹 ∩ 𝒫 𝑥)) | |
6 | 5 | neeq1d 2853 | . . . . 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 3195 | . . 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 6191 | . . 3 ⊢ (𝑣 = 𝑋 → (fBas‘𝑣) = (fBas‘𝑋)) | |
11 | 10 | adantl 482 | . 2 ⊢ ((𝐹 ∈ (fBas‘𝑋) ∧ 𝑣 = 𝑋) → (fBas‘𝑣) = (fBas‘𝑋)) |
12 | elfvex 6221 | . 2 ⊢ (𝐹 ∈ (fBas‘𝑋) → 𝑋 ∈ V) | |
13 | id 22 | . 2 ⊢ (𝐹 ∈ (fBas‘𝑋) → 𝐹 ∈ (fBas‘𝑋)) | |
14 | elfvdm 6220 | . . 3 ⊢ (𝐹 ∈ (fBas‘𝑋) → 𝑋 ∈ dom fBas) | |
15 | pwexg 4850 | . . 3 ⊢ (𝑋 ∈ dom fBas → 𝒫 𝑋 ∈ V) | |
16 | rabexg 4812 | . . 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 6787 | 1 ⊢ (𝐹 ∈ (fBas‘𝑋) → (𝑋filGen𝐹) = {𝑥 ∈ 𝒫 𝑋 ∣ (𝐹 ∩ 𝒫 𝑥) ≠ ∅}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ≠ wne 2794 {crab 2916 Vcvv 3200 ∩ cin 3573 ∅c0 3915 𝒫 cpw 4158 dom cdm 5114 ‘cfv 5888 (class class class)co 6650 ↦ cmpt2 6652 fBascfbas 19734 filGencfg 19735 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-iota 5851 df-fun 5890 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-fg 19744 |
This theorem is referenced by: elfg 21675 restmetu 22375 neifg 32366 |
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