Theorem filunirn 23803

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 6841	. . . . . 6 ⊢ (fBas‘𝑦) ∈ V
2	1	rabex 5279	. . . . 5 ⊢ {𝑤 ∈ (fBas‘𝑦) ∣ ∀𝑧 ∈ 𝒫 𝑦((𝑤 ∩ 𝒫 𝑧) ≠ ∅ → 𝑧 ∈ 𝑤)} ∈ V
3		df-fil 23767	. . . . 5 ⊢ Fil = (𝑦 ∈ V ↦ {𝑤 ∈ (fBas‘𝑦) ∣ ∀𝑧 ∈ 𝒫 𝑦((𝑤 ∩ 𝒫 𝑧) ≠ ∅ → 𝑧 ∈ 𝑤)})
4	2, 3	fnmpti 6630	. . . 4 ⊢ Fil Fn V
5		fnunirn 7193	. . . 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 23802	. . . . . . 7 ⊢ (𝐹 ∈ (Fil‘𝑥) → ∪ 𝐹 = 𝑥)
8	7	fveq2d 6832	. . . . . 6 ⊢ (𝐹 ∈ (Fil‘𝑥) → (Fil‘∪ 𝐹) = (Fil‘𝑥))
9	8	eleq2d 2817	. . . . 5 ⊢ (𝐹 ∈ (Fil‘𝑥) → (𝐹 ∈ (Fil‘∪ 𝐹) ↔ 𝐹 ∈ (Fil‘𝑥)))
10	9	ibir 268	. . . 4 ⊢ (𝐹 ∈ (Fil‘𝑥) → 𝐹 ∈ (Fil‘∪ 𝐹))
11	10	rexlimivw 3129	. . 3 ⊢ (∃𝑥 ∈ V 𝐹 ∈ (Fil‘𝑥) → 𝐹 ∈ (Fil‘∪ 𝐹))
12	6, 11	sylbi 217	. 2 ⊢ (𝐹 ∈ ∪ ran Fil → 𝐹 ∈ (Fil‘∪ 𝐹))
13		fvssunirn 6859	. . 3 ⊢ (Fil‘∪ 𝐹) ⊆ ∪ ran Fil
14	13	sseli 3925	. 2 ⊢ (𝐹 ∈ (Fil‘∪ 𝐹) → 𝐹 ∈ ∪ ran Fil)
15	12, 14	impbii 209	1 ⊢ (𝐹 ∈ ∪ ran Fil ↔ 𝐹 ∈ (Fil‘∪ 𝐹))

Syntax hints:   → wi 4   ↔ wb 206   ∈ wcel 2111   ≠ wne 2928  ∀wral 3047  ∃wrex 3056  {crab 3395  Vcvv 3436   ∩ cin 3896  ∅c0 4282  𝒫 cpw 4549  ∪ cuni 4858  ran crn 5620   Fn wfn 6482  ‘cfv 6487  fBascfbas 21285  Filcfil 23766

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 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5236  ax-nul 5246  ax-pow 5305  ax-pr 5372

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-nel 3033  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4283  df-if 4475  df-pw 4551  df-sn 4576  df-pr 4578  df-op 4582  df-uni 4859  df-br 5094  df-opab 5156  df-mpt 5175  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-iota 6443  df-fun 6489  df-fn 6490  df-fv 6495  df-fbas 21294  df-fil 23767

This theorem is referenced by:  flimfil  23890  isfcls  23930