Theorem filunirn 23868

Description: Two ways to express a filter on an unspecified base. (Contributed by Stefan O'Rear, 2-Aug-2015.)

Assertion

Ref	Expression
filunirn	⊢ (𝐹 ∈ ∪ ran Fil ↔ 𝐹 ∈ (Fil‘∪ 𝐹))

Proof of Theorem filunirn

Dummy variables 𝑦 𝑤 𝑧 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fvex 6843	. . . . . 6 ⊢ (fBas‘𝑦) ∈ V
2	1	rabex 5270	. . . . 5 ⊢ {𝑤 ∈ (fBas‘𝑦) ∣ ∀𝑧 ∈ 𝒫 𝑦((𝑤 ∩ 𝒫 𝑧) ≠ ∅ → 𝑧 ∈ 𝑤)} ∈ V
3		df-fil 23832	. . . . 5 ⊢ Fil = (𝑦 ∈ V ↦ {𝑤 ∈ (fBas‘𝑦) ∣ ∀𝑧 ∈ 𝒫 𝑦((𝑤 ∩ 𝒫 𝑧) ≠ ∅ → 𝑧 ∈ 𝑤)})
4	2, 3	fnmpti 6631	. . . 4 ⊢ Fil Fn V
5		fnunirn 7200	. . . 4 ⊢ (Fil Fn V → (𝐹 ∈ ∪ ran Fil ↔ ∃𝑥 ∈ V 𝐹 ∈ (Fil‘𝑥)))
6	4, 5	ax-mp 5	. . 3 ⊢ (𝐹 ∈ ∪ ran Fil ↔ ∃𝑥 ∈ V 𝐹 ∈ (Fil‘𝑥))
7		filunibas 23867	. . . . . . 7 ⊢ (𝐹 ∈ (Fil‘𝑥) → ∪ 𝐹 = 𝑥)
8	7	fveq2d 6834	. . . . . 6 ⊢ (𝐹 ∈ (Fil‘𝑥) → (Fil‘∪ 𝐹) = (Fil‘𝑥))
9	8	eleq2d 2822	. . . . 5 ⊢ (𝐹 ∈ (Fil‘𝑥) → (𝐹 ∈ (Fil‘∪ 𝐹) ↔ 𝐹 ∈ (Fil‘𝑥)))
10	9	ibir 269	. . . 4 ⊢ (𝐹 ∈ (Fil‘𝑥) → 𝐹 ∈ (Fil‘∪ 𝐹))
11	10	rexlimivw 3133	. . 3 ⊢ (∃𝑥 ∈ V 𝐹 ∈ (Fil‘𝑥) → 𝐹 ∈ (Fil‘∪ 𝐹))
12	6, 11	sylbi 218	. 2 ⊢ (𝐹 ∈ ∪ ran Fil → 𝐹 ∈ (Fil‘∪ 𝐹))
13		fvssunirn 6861	. . 3 ⊢ (Fil‘∪ 𝐹) ⊆ ∪ ran Fil
14	13	sseli 3914	. 2 ⊢ (𝐹 ∈ (Fil‘∪ 𝐹) → 𝐹 ∈ ∪ ran Fil)
15	12, 14	impbii 210	1 ⊢ (𝐹 ∈ ∪ ran Fil ↔ 𝐹 ∈ (Fil‘∪ 𝐹))

Syntax hints:   → wi 4   ↔ wb 207   ∈ wcel 2115   ≠ wne 2931  ∀wral 3050  ∃wrex 3060  {crab 3388  Vcvv 3428   ∩ cin 3885  ∅c0 4264  𝒫 cpw 4532  ∪ cuni 4841  ran crn 5622   Fn wfn 6483  ‘cfv 6488  fBascfbas 21338  Filcfil 23831

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1970  ax-7 2011  ax-8 2117  ax-9 2125  ax-10 2148  ax-11 2164  ax-12 2185  ax-ext 2708  ax-sep 5221  ax-nul 5231  ax-pow 5297  ax-pr 5365

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 850  df-3an 1090  df-tru 1546  df-fal 1556  df-ex 1783  df-nf 1787  df-sb 2070  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2932  df-nel 3036  df-ral 3051  df-rex 3061  df-rab 3389  df-v 3430  df-sbc 3727  df-csb 3835  df-dif 3889  df-un 3891  df-in 3893  df-ss 3903  df-nul 4265  df-if 4458  df-pw 4534  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4842  df-br 5076  df-opab 5138  df-mpt 5157  df-id 5516  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-iota 6444  df-fun 6490  df-fn 6491  df-fv 6496  df-fbas 21347  df-fil 23832

This theorem is referenced by:  flimfil  23955  isfcls  23995