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Mirrors > Home > MPE Home > Th. List > elfg | Structured version Visualization version GIF version |
Description: A condition for elements of a generated filter. (Contributed by Jeff Hankins, 3-Sep-2009.) (Revised by Stefan O'Rear, 2-Aug-2015.) |
Ref | Expression |
---|---|
elfg | ⊢ (𝐹 ∈ (fBas‘𝑋) → (𝐴 ∈ (𝑋filGen𝐹) ↔ (𝐴 ⊆ 𝑋 ∧ ∃𝑥 ∈ 𝐹 𝑥 ⊆ 𝐴))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fgval 22478 | . . 3 ⊢ (𝐹 ∈ (fBas‘𝑋) → (𝑋filGen𝐹) = {𝑦 ∈ 𝒫 𝑋 ∣ (𝐹 ∩ 𝒫 𝑦) ≠ ∅}) | |
2 | 1 | eleq2d 2898 | . 2 ⊢ (𝐹 ∈ (fBas‘𝑋) → (𝐴 ∈ (𝑋filGen𝐹) ↔ 𝐴 ∈ {𝑦 ∈ 𝒫 𝑋 ∣ (𝐹 ∩ 𝒫 𝑦) ≠ ∅})) |
3 | pweq 4555 | . . . . . 6 ⊢ (𝑦 = 𝐴 → 𝒫 𝑦 = 𝒫 𝐴) | |
4 | 3 | ineq2d 4189 | . . . . 5 ⊢ (𝑦 = 𝐴 → (𝐹 ∩ 𝒫 𝑦) = (𝐹 ∩ 𝒫 𝐴)) |
5 | 4 | neeq1d 3075 | . . . 4 ⊢ (𝑦 = 𝐴 → ((𝐹 ∩ 𝒫 𝑦) ≠ ∅ ↔ (𝐹 ∩ 𝒫 𝐴) ≠ ∅)) |
6 | 5 | elrab 3680 | . . 3 ⊢ (𝐴 ∈ {𝑦 ∈ 𝒫 𝑋 ∣ (𝐹 ∩ 𝒫 𝑦) ≠ ∅} ↔ (𝐴 ∈ 𝒫 𝑋 ∧ (𝐹 ∩ 𝒫 𝐴) ≠ ∅)) |
7 | elfvdm 6702 | . . . . 5 ⊢ (𝐹 ∈ (fBas‘𝑋) → 𝑋 ∈ dom fBas) | |
8 | elpw2g 5247 | . . . . 5 ⊢ (𝑋 ∈ dom fBas → (𝐴 ∈ 𝒫 𝑋 ↔ 𝐴 ⊆ 𝑋)) | |
9 | 7, 8 | syl 17 | . . . 4 ⊢ (𝐹 ∈ (fBas‘𝑋) → (𝐴 ∈ 𝒫 𝑋 ↔ 𝐴 ⊆ 𝑋)) |
10 | elin 4169 | . . . . . . . 8 ⊢ (𝑥 ∈ (𝐹 ∩ 𝒫 𝐴) ↔ (𝑥 ∈ 𝐹 ∧ 𝑥 ∈ 𝒫 𝐴)) | |
11 | velpw 4544 | . . . . . . . . 9 ⊢ (𝑥 ∈ 𝒫 𝐴 ↔ 𝑥 ⊆ 𝐴) | |
12 | 11 | anbi2i 624 | . . . . . . . 8 ⊢ ((𝑥 ∈ 𝐹 ∧ 𝑥 ∈ 𝒫 𝐴) ↔ (𝑥 ∈ 𝐹 ∧ 𝑥 ⊆ 𝐴)) |
13 | 10, 12 | bitri 277 | . . . . . . 7 ⊢ (𝑥 ∈ (𝐹 ∩ 𝒫 𝐴) ↔ (𝑥 ∈ 𝐹 ∧ 𝑥 ⊆ 𝐴)) |
14 | 13 | exbii 1848 | . . . . . 6 ⊢ (∃𝑥 𝑥 ∈ (𝐹 ∩ 𝒫 𝐴) ↔ ∃𝑥(𝑥 ∈ 𝐹 ∧ 𝑥 ⊆ 𝐴)) |
15 | n0 4310 | . . . . . 6 ⊢ ((𝐹 ∩ 𝒫 𝐴) ≠ ∅ ↔ ∃𝑥 𝑥 ∈ (𝐹 ∩ 𝒫 𝐴)) | |
16 | df-rex 3144 | . . . . . 6 ⊢ (∃𝑥 ∈ 𝐹 𝑥 ⊆ 𝐴 ↔ ∃𝑥(𝑥 ∈ 𝐹 ∧ 𝑥 ⊆ 𝐴)) | |
17 | 14, 15, 16 | 3bitr4i 305 | . . . . 5 ⊢ ((𝐹 ∩ 𝒫 𝐴) ≠ ∅ ↔ ∃𝑥 ∈ 𝐹 𝑥 ⊆ 𝐴) |
18 | 17 | a1i 11 | . . . 4 ⊢ (𝐹 ∈ (fBas‘𝑋) → ((𝐹 ∩ 𝒫 𝐴) ≠ ∅ ↔ ∃𝑥 ∈ 𝐹 𝑥 ⊆ 𝐴)) |
19 | 9, 18 | anbi12d 632 | . . 3 ⊢ (𝐹 ∈ (fBas‘𝑋) → ((𝐴 ∈ 𝒫 𝑋 ∧ (𝐹 ∩ 𝒫 𝐴) ≠ ∅) ↔ (𝐴 ⊆ 𝑋 ∧ ∃𝑥 ∈ 𝐹 𝑥 ⊆ 𝐴))) |
20 | 6, 19 | syl5bb 285 | . 2 ⊢ (𝐹 ∈ (fBas‘𝑋) → (𝐴 ∈ {𝑦 ∈ 𝒫 𝑋 ∣ (𝐹 ∩ 𝒫 𝑦) ≠ ∅} ↔ (𝐴 ⊆ 𝑋 ∧ ∃𝑥 ∈ 𝐹 𝑥 ⊆ 𝐴))) |
21 | 2, 20 | bitrd 281 | 1 ⊢ (𝐹 ∈ (fBas‘𝑋) → (𝐴 ∈ (𝑋filGen𝐹) ↔ (𝐴 ⊆ 𝑋 ∧ ∃𝑥 ∈ 𝐹 𝑥 ⊆ 𝐴))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1537 ∃wex 1780 ∈ wcel 2114 ≠ wne 3016 ∃wrex 3139 {crab 3142 ∩ cin 3935 ⊆ wss 3936 ∅c0 4291 𝒫 cpw 4539 dom cdm 5555 ‘cfv 6355 (class class class)co 7156 fBascfbas 20533 filGencfg 20534 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-iota 6314 df-fun 6357 df-fv 6363 df-ov 7159 df-oprab 7160 df-mpo 7161 df-fg 20543 |
This theorem is referenced by: ssfg 22480 fgss 22481 fgss2 22482 fgfil 22483 elfilss 22484 fgcl 22486 fgabs 22487 fgtr 22498 trfg 22499 uffix 22529 elfm 22555 elfm2 22556 elfm3 22558 fbflim 22584 flffbas 22603 fclsbas 22629 isucn2 22888 metust 23168 cfilucfil 23169 metuel 23174 fgcfil 23874 fgmin 33718 filnetlem4 33729 |
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