Theorem filunirn 24050

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 6894	. . . . . 6 ⊢ (fBas‘𝑦) ∈ V
2	1	rabex 5308	. . . . 5 ⊢ {𝑤 ∈ (fBas‘𝑦) ∣ ∀𝑧 ∈ 𝒫 𝑦((𝑤 ∩ 𝒫 𝑧) ≠ ∅ → 𝑧 ∈ 𝑤)} ∈ V
3		df-fil 24014	. . . . 5 ⊢ Fil = (𝑦 ∈ V ↦ {𝑤 ∈ (fBas‘𝑦) ∣ ∀𝑧 ∈ 𝒫 𝑦((𝑤 ∩ 𝒫 𝑧) ≠ ∅ → 𝑧 ∈ 𝑤)})
4	2, 3	fnmpti 6678	. . . 4 ⊢ Fil Fn V
5		fnunirn 7251	. . . 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 24049	. . . . . . 7 ⊢ (𝐹 ∈ (Fil‘𝑥) → ∪ 𝐹 = 𝑥)
8	7	fveq2d 6885	. . . . . 6 ⊢ (𝐹 ∈ (Fil‘𝑥) → (Fil‘∪ 𝐹) = (Fil‘𝑥))
9	8	eleq2d 2848	. . . . 5 ⊢ (𝐹 ∈ (Fil‘𝑥) → (𝐹 ∈ (Fil‘∪ 𝐹) ↔ 𝐹 ∈ (Fil‘𝑥)))
10	9	ibir 271	. . . 4 ⊢ (𝐹 ∈ (Fil‘𝑥) → 𝐹 ∈ (Fil‘∪ 𝐹))
11	10	rexlimivw 3161	. . 3 ⊢ (∃𝑥 ∈ V 𝐹 ∈ (Fil‘𝑥) → 𝐹 ∈ (Fil‘∪ 𝐹))
12	6, 11	sylbi 220	. 2 ⊢ (𝐹 ∈ ∪ ran Fil → 𝐹 ∈ (Fil‘∪ 𝐹))
13		fvssunirn 6912	. . 3 ⊢ (Fil‘∪ 𝐹) ⊆ ∪ ran Fil
14	13	sseli 3932	. 2 ⊢ (𝐹 ∈ (Fil‘∪ 𝐹) → 𝐹 ∈ ∪ ran Fil)
15	12, 14	impbii 212	1 ⊢ (𝐹 ∈ ∪ ran Fil ↔ 𝐹 ∈ (Fil‘∪ 𝐹))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∈ wcel 2142   ≠ wne 2957  ∀wral 3078  ∃wrex 3088  {crab 3415  Vcvv 3454   ∩ cin 3903  ∅c0 4285  𝒫 cpw 4561  ∪ cuni 4871  ran crn 5661   Fn wfn 6531  ‘cfv 6536  fBascfbas 21521  Filcfil 24013

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-sep 5256  ax-nul 5268  ax-pow 5335  ax-pr 5403

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-nf 1813  df-sb 2096  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-nel 3064  df-ral 3079  df-rex 3089  df-rab 3416  df-v 3456  df-sbc 3744  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-pw 4563  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-br 5109  df-opab 5173  df-mpt 5192  df-id 5555  df-xp 5666  df-rel 5667  df-cnv 5668  df-co 5669  df-dm 5670  df-rn 5671  df-res 5672  df-ima 5673  df-iota 6492  df-fun 6538  df-fn 6539  df-fv 6544  df-fbas 21530  df-fil 24014

This theorem is used by:  flimfil  24137  isfcls  24177