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Theorem fgval 22406
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.)
Assertion
Ref Expression
fgval (𝐹 ∈ (fBas‘𝑋) → (𝑋filGen𝐹) = {𝑥 ∈ 𝒫 𝑋 ∣ (𝐹 ∩ 𝒫 𝑥) ≠ ∅})
Distinct variable groups:   𝑥,𝐹   𝑥,𝑋

Proof of Theorem fgval
Dummy variables 𝑣 𝑓 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-fg 20471 . . 3 filGen = (𝑣 ∈ V, 𝑓 ∈ (fBas‘𝑣) ↦ {𝑥 ∈ 𝒫 𝑣 ∣ (𝑓 ∩ 𝒫 𝑥) ≠ ∅})
21a1i 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)
43adantr 481 . . . 4 ((𝑣 = 𝑋𝑓 = 𝐹) → 𝒫 𝑣 = 𝒫 𝑋)
5 ineq1 4178 . . . . . 6 (𝑓 = 𝐹 → (𝑓 ∩ 𝒫 𝑥) = (𝐹 ∩ 𝒫 𝑥))
65neeq1d 3072 . . . . 5 (𝑓 = 𝐹 → ((𝑓 ∩ 𝒫 𝑥) ≠ ∅ ↔ (𝐹 ∩ 𝒫 𝑥) ≠ ∅))
76adantl 482 . . . 4 ((𝑣 = 𝑋𝑓 = 𝐹) → ((𝑓 ∩ 𝒫 𝑥) ≠ ∅ ↔ (𝐹 ∩ 𝒫 𝑥) ≠ ∅))
84, 7rabeqbidv 3483 . . 3 ((𝑣 = 𝑋𝑓 = 𝐹) → {𝑥 ∈ 𝒫 𝑣 ∣ (𝑓 ∩ 𝒫 𝑥) ≠ ∅} = {𝑥 ∈ 𝒫 𝑋 ∣ (𝐹 ∩ 𝒫 𝑥) ≠ ∅})
98adantl 482 . 2 ((𝐹 ∈ (fBas‘𝑋) ∧ (𝑣 = 𝑋𝑓 = 𝐹)) → {𝑥 ∈ 𝒫 𝑣 ∣ (𝑓 ∩ 𝒫 𝑥) ≠ ∅} = {𝑥 ∈ 𝒫 𝑋 ∣ (𝐹 ∩ 𝒫 𝑥) ≠ ∅})
10 fveq2 6663 . . 3 (𝑣 = 𝑋 → (fBas‘𝑣) = (fBas‘𝑋))
1110adantl 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)
1714, 15, 163syl 18 . 2 (𝐹 ∈ (fBas‘𝑋) → {𝑥 ∈ 𝒫 𝑋 ∣ (𝐹 ∩ 𝒫 𝑥) ≠ ∅} ∈ V)
182, 9, 11, 12, 13, 17ovmpodx 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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