Theorem fgval 23835

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.

Step	Hyp	Ref	Expression
1		df-fg 21350	. . 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 4555	. . . . 5 ⊢ (𝑣 = 𝑋 → 𝒫 𝑣 = 𝒫 𝑋)
4	3	adantr 480	. . . 4 ⊢ ((𝑣 = 𝑋 ∧ 𝑓 = 𝐹) → 𝒫 𝑣 = 𝒫 𝑋)
5		ineq1 4153	. . . . . 6 ⊢ (𝑓 = 𝐹 → (𝑓 ∩ 𝒫 𝑥) = (𝐹 ∩ 𝒫 𝑥))
6	5	neeq1d 2991	. . . . 5 ⊢ (𝑓 = 𝐹 → ((𝑓 ∩ 𝒫 𝑥) ≠ ∅ ↔ (𝐹 ∩ 𝒫 𝑥) ≠ ∅))
7	6	adantl 481	. . . 4 ⊢ ((𝑣 = 𝑋 ∧ 𝑓 = 𝐹) → ((𝑓 ∩ 𝒫 𝑥) ≠ ∅ ↔ (𝐹 ∩ 𝒫 𝑥) ≠ ∅))
8	4, 7	rabeqbidv 3407	. . 3 ⊢ ((𝑣 = 𝑋 ∧ 𝑓 = 𝐹) → {𝑥 ∈ 𝒫 𝑣 ∣ (𝑓 ∩ 𝒫 𝑥) ≠ ∅} = {𝑥 ∈ 𝒫 𝑋 ∣ (𝐹 ∩ 𝒫 𝑥) ≠ ∅})
9	8	adantl 481	. 2 ⊢ ((𝐹 ∈ (fBas‘𝑋) ∧ (𝑣 = 𝑋 ∧ 𝑓 = 𝐹)) → {𝑥 ∈ 𝒫 𝑣 ∣ (𝑓 ∩ 𝒫 𝑥) ≠ ∅} = {𝑥 ∈ 𝒫 𝑋 ∣ (𝐹 ∩ 𝒫 𝑥) ≠ ∅})
10		fveq2 6840	. . 3 ⊢ (𝑣 = 𝑋 → (fBas‘𝑣) = (fBas‘𝑋))
11	10	adantl 481	. 2 ⊢ ((𝐹 ∈ (fBas‘𝑋) ∧ 𝑣 = 𝑋) → (fBas‘𝑣) = (fBas‘𝑋))
12		elfvex 6875	. 2 ⊢ (𝐹 ∈ (fBas‘𝑋) → 𝑋 ∈ V)
13		id 22	. 2 ⊢ (𝐹 ∈ (fBas‘𝑋) → 𝐹 ∈ (fBas‘𝑋))
14		elfvdm 6874	. . 3 ⊢ (𝐹 ∈ (fBas‘𝑋) → 𝑋 ∈ dom fBas)
15		pwexg 5320	. . 3 ⊢ (𝑋 ∈ dom fBas → 𝒫 𝑋 ∈ V)
16		rabexg 5278	. . 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	ovmpodx 7518	1 ⊢ (𝐹 ∈ (fBas‘𝑋) → (𝑋filGen𝐹) = {𝑥 ∈ 𝒫 𝑋 ∣ (𝐹 ∩ 𝒫 𝑥) ≠ ∅})

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114   ≠ wne 2932  {crab 3389  Vcvv 3429   ∩ cin 3888  ∅c0 4273  𝒫 cpw 4541  dom cdm 5631  ‘cfv 6498  (class class class)co 7367   ∈ cmpo 7369  fBascfbas 21340  filGencfg 21341

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-rab 3390  df-v 3431  df-sbc 3729  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-br 5086  df-opab 5148  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-iota 6454  df-fun 6500  df-fv 6506  df-ov 7370  df-oprab 7371  df-mpo 7372  df-fg 21350

This theorem is referenced by:  elfg  23836  restmetu  24535  neifg  36553