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Theorem fgval 22478
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 20092 . . 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 4516 . . . . 5 (𝑣 = 𝑋 → 𝒫 𝑣 = 𝒫 𝑋)
43adantr 484 . . . 4 ((𝑣 = 𝑋𝑓 = 𝐹) → 𝒫 𝑣 = 𝒫 𝑋)
5 ineq1 4134 . . . . . 6 (𝑓 = 𝐹 → (𝑓 ∩ 𝒫 𝑥) = (𝐹 ∩ 𝒫 𝑥))
65neeq1d 3049 . . . . 5 (𝑓 = 𝐹 → ((𝑓 ∩ 𝒫 𝑥) ≠ ∅ ↔ (𝐹 ∩ 𝒫 𝑥) ≠ ∅))
76adantl 485 . . . 4 ((𝑣 = 𝑋𝑓 = 𝐹) → ((𝑓 ∩ 𝒫 𝑥) ≠ ∅ ↔ (𝐹 ∩ 𝒫 𝑥) ≠ ∅))
84, 7rabeqbidv 3436 . . 3 ((𝑣 = 𝑋𝑓 = 𝐹) → {𝑥 ∈ 𝒫 𝑣 ∣ (𝑓 ∩ 𝒫 𝑥) ≠ ∅} = {𝑥 ∈ 𝒫 𝑋 ∣ (𝐹 ∩ 𝒫 𝑥) ≠ ∅})
98adantl 485 . 2 ((𝐹 ∈ (fBas‘𝑋) ∧ (𝑣 = 𝑋𝑓 = 𝐹)) → {𝑥 ∈ 𝒫 𝑣 ∣ (𝑓 ∩ 𝒫 𝑥) ≠ ∅} = {𝑥 ∈ 𝒫 𝑋 ∣ (𝐹 ∩ 𝒫 𝑥) ≠ ∅})
10 fveq2 6649 . . 3 (𝑣 = 𝑋 → (fBas‘𝑣) = (fBas‘𝑋))
1110adantl 485 . 2 ((𝐹 ∈ (fBas‘𝑋) ∧ 𝑣 = 𝑋) → (fBas‘𝑣) = (fBas‘𝑋))
12 elfvex 6682 . 2 (𝐹 ∈ (fBas‘𝑋) → 𝑋 ∈ V)
13 id 22 . 2 (𝐹 ∈ (fBas‘𝑋) → 𝐹 ∈ (fBas‘𝑋))
14 elfvdm 6681 . . 3 (𝐹 ∈ (fBas‘𝑋) → 𝑋 ∈ dom fBas)
15 pwexg 5247 . . 3 (𝑋 ∈ dom fBas → 𝒫 𝑋 ∈ V)
16 rabexg 5201 . . 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 7284 1 (𝐹 ∈ (fBas‘𝑋) → (𝑋filGen𝐹) = {𝑥 ∈ 𝒫 𝑋 ∣ (𝐹 ∩ 𝒫 𝑥) ≠ ∅})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1538  wcel 2112  wne 2990  {crab 3113  Vcvv 3444  cin 3883  c0 4246  𝒫 cpw 4500  dom cdm 5523  cfv 6328  (class class class)co 7139  cmpo 7141  fBascfbas 20082  filGencfg 20083
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-sbc 3724  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-br 5034  df-opab 5096  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-iota 6287  df-fun 6330  df-fv 6336  df-ov 7142  df-oprab 7143  df-mpo 7144  df-fg 20092
This theorem is referenced by:  elfg  22479  restmetu  23180  neifg  33827
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