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.

Step	Hyp	Ref	Expression
1		df-fg 20543	. . 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 483	. . . 4 ⊢ ((𝑣 = 𝑋 ∧ 𝑓 = 𝐹) → 𝒫 𝑣 = 𝒫 𝑋)
5		ineq1 4181	. . . . . 6 ⊢ (𝑓 = 𝐹 → (𝑓 ∩ 𝒫 𝑥) = (𝐹 ∩ 𝒫 𝑥))
6	5	neeq1d 3075	. . . . 5 ⊢ (𝑓 = 𝐹 → ((𝑓 ∩ 𝒫 𝑥) ≠ ∅ ↔ (𝐹 ∩ 𝒫 𝑥) ≠ ∅))
7	6	adantl 484	. . . 4 ⊢ ((𝑣 = 𝑋 ∧ 𝑓 = 𝐹) → ((𝑓 ∩ 𝒫 𝑥) ≠ ∅ ↔ (𝐹 ∩ 𝒫 𝑥) ≠ ∅))
8	4, 7	rabeqbidv 3485	. . 3 ⊢ ((𝑣 = 𝑋 ∧ 𝑓 = 𝐹) → {𝑥 ∈ 𝒫 𝑣 ∣ (𝑓 ∩ 𝒫 𝑥) ≠ ∅} = {𝑥 ∈ 𝒫 𝑋 ∣ (𝐹 ∩ 𝒫 𝑥) ≠ ∅})
9	8	adantl 484	. 2 ⊢ ((𝐹 ∈ (fBas‘𝑋) ∧ (𝑣 = 𝑋 ∧ 𝑓 = 𝐹)) → {𝑥 ∈ 𝒫 𝑣 ∣ (𝑓 ∩ 𝒫 𝑥) ≠ ∅} = {𝑥 ∈ 𝒫 𝑋 ∣ (𝐹 ∩ 𝒫 𝑥) ≠ ∅})
10		fveq2 6670	. . 3 ⊢ (𝑣 = 𝑋 → (fBas‘𝑣) = (fBas‘𝑋))
11	10	adantl 484	. 2 ⊢ ((𝐹 ∈ (fBas‘𝑋) ∧ 𝑣 = 𝑋) → (fBas‘𝑣) = (fBas‘𝑋))
12		elfvex 6703	. 2 ⊢ (𝐹 ∈ (fBas‘𝑋) → 𝑋 ∈ V)
13		id 22	. 2 ⊢ (𝐹 ∈ (fBas‘𝑋) → 𝐹 ∈ (fBas‘𝑋))
14		elfvdm 6702	. . 3 ⊢ (𝐹 ∈ (fBas‘𝑋) → 𝑋 ∈ dom fBas)
15		pwexg 5279	. . 3 ⊢ (𝑋 ∈ dom fBas → 𝒫 𝑋 ∈ V)
16		rabexg 5234	. . 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 7301	1 ⊢ (𝐹 ∈ (fBas‘𝑋) → (𝑋filGen𝐹) = {𝑥 ∈ 𝒫 𝑋 ∣ (𝐹 ∩ 𝒫 𝑥) ≠ ∅})

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1537   ∈ wcel 2114   ≠ wne 3016  {crab 3142  Vcvv 3494   ∩ cin 3935  ∅c0 4291  𝒫 cpw 4539  dom cdm 5555  ‘cfv 6355  (class class class)co 7156   ∈ cmpo 7158  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:  elfg  22479  restmetu  23180  neifg  33719