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Theorem fgval 21579
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 19658 . . 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 4138 . . . . 5 (𝑣 = 𝑋 → 𝒫 𝑣 = 𝒫 𝑋)
43adantr 481 . . . 4 ((𝑣 = 𝑋𝑓 = 𝐹) → 𝒫 𝑣 = 𝒫 𝑋)
5 ineq1 3790 . . . . . 6 (𝑓 = 𝐹 → (𝑓 ∩ 𝒫 𝑥) = (𝐹 ∩ 𝒫 𝑥))
65neeq1d 2855 . . . . 5 (𝑓 = 𝐹 → ((𝑓 ∩ 𝒫 𝑥) ≠ ∅ ↔ (𝐹 ∩ 𝒫 𝑥) ≠ ∅))
76adantl 482 . . . 4 ((𝑣 = 𝑋𝑓 = 𝐹) → ((𝑓 ∩ 𝒫 𝑥) ≠ ∅ ↔ (𝐹 ∩ 𝒫 𝑥) ≠ ∅))
84, 7rabeqbidv 3186 . . 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 6150 . . 3 (𝑣 = 𝑋 → (fBas‘𝑣) = (fBas‘𝑋))
1110adantl 482 . 2 ((𝐹 ∈ (fBas‘𝑋) ∧ 𝑣 = 𝑋) → (fBas‘𝑣) = (fBas‘𝑋))
12 elfvex 6179 . 2 (𝐹 ∈ (fBas‘𝑋) → 𝑋 ∈ V)
13 id 22 . 2 (𝐹 ∈ (fBas‘𝑋) → 𝐹 ∈ (fBas‘𝑋))
14 elfvdm 6178 . . 3 (𝐹 ∈ (fBas‘𝑋) → 𝑋 ∈ dom fBas)
15 pwexg 4815 . . 3 (𝑋 ∈ dom fBas → 𝒫 𝑋 ∈ V)
16 rabexg 4777 . . 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, 17ovmpt2dx 6741 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 1480  wcel 1992  wne 2796  {crab 2916  Vcvv 3191  cin 3559  c0 3896  𝒫 cpw 4135  dom cdm 5079  cfv 5850  (class class class)co 6605  cmpt2 6607  fBascfbas 19648  filGencfg 19649
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3193  df-sbc 3423  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-br 4619  df-opab 4679  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-iota 5813  df-fun 5852  df-fv 5858  df-ov 6608  df-oprab 6609  df-mpt2 6610  df-fg 19658
This theorem is referenced by:  elfg  21580  restmetu  22280  neifg  32000
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