Theorem flimtopon 24197

Description: Reverse closure for the limit point predicate. (Contributed by Mario Carneiro, 26-Aug-2015.)

Assertion

Ref	Expression
flimtopon	⊢ (𝐴 ∈ (𝐽 fLim 𝐹) → (𝐽 ∈ (TopOn‘𝑋) ↔ 𝐹 ∈ (Fil‘𝑋)))

Proof of Theorem flimtopon

Step	Hyp	Ref	Expression
1		flimtop 24192	. . 3 ⊢ (𝐴 ∈ (𝐽 fLim 𝐹) → 𝐽 ∈ Top)
2		istopon 23138	. . . 4 ⊢ (𝐽 ∈ (TopOn‘𝑋) ↔ (𝐽 ∈ Top ∧ 𝑋 = ∪ 𝐽))
3	2	baib 545	. . 3 ⊢ (𝐽 ∈ Top → (𝐽 ∈ (TopOn‘𝑋) ↔ 𝑋 = ∪ 𝐽))
4	1, 3	syl 18	. 2 ⊢ (𝐴 ∈ (𝐽 fLim 𝐹) → (𝐽 ∈ (TopOn‘𝑋) ↔ 𝑋 = ∪ 𝐽))
5		eqid 2760	. . . . 5 ⊢ ∪ 𝐽 = ∪ 𝐽
6	5	flimfil 24196	. . . 4 ⊢ (𝐴 ∈ (𝐽 fLim 𝐹) → 𝐹 ∈ (Fil‘∪ 𝐽))
7		fveq2 6879	. . . . 5 ⊢ (𝑋 = ∪ 𝐽 → (Fil‘𝑋) = (Fil‘∪ 𝐽))
8	7	eleq2d 2846	. . . 4 ⊢ (𝑋 = ∪ 𝐽 → (𝐹 ∈ (Fil‘𝑋) ↔ 𝐹 ∈ (Fil‘∪ 𝐽)))
9	6, 8	syl5ibrcom 250	. . 3 ⊢ (𝐴 ∈ (𝐽 fLim 𝐹) → (𝑋 = ∪ 𝐽 → 𝐹 ∈ (Fil‘𝑋)))
10		filunibas 24108	. . . . 5 ⊢ (𝐹 ∈ (Fil‘∪ 𝐽) → ∪ 𝐹 = ∪ 𝐽)
11	6, 10	syl 18	. . . 4 ⊢ (𝐴 ∈ (𝐽 fLim 𝐹) → ∪ 𝐹 = ∪ 𝐽)
12		filunibas 24108	. . . . 5 ⊢ (𝐹 ∈ (Fil‘𝑋) → ∪ 𝐹 = 𝑋)
13	12	eqeq1d 2762	. . . 4 ⊢ (𝐹 ∈ (Fil‘𝑋) → (∪ 𝐹 = ∪ 𝐽 ↔ 𝑋 = ∪ 𝐽))
14	11, 13	syl5ibcom 248	. . 3 ⊢ (𝐴 ∈ (𝐽 fLim 𝐹) → (𝐹 ∈ (Fil‘𝑋) → 𝑋 = ∪ 𝐽))
15	9, 14	impbid 215	. 2 ⊢ (𝐴 ∈ (𝐽 fLim 𝐹) → (𝑋 = ∪ 𝐽 ↔ 𝐹 ∈ (Fil‘𝑋)))
16	4, 15	bitrd 282	1 ⊢ (𝐴 ∈ (𝐽 fLim 𝐹) → (𝐽 ∈ (TopOn‘𝑋) ↔ 𝐹 ∈ (Fil‘𝑋)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   = wceq 1570   ∈ wcel 2145  ∪ cuni 4867  ‘cfv 6533  (class class class)co 7414  Topctop 23119  TopOnctopon 23136  Filcfil 24072   fLim cflim 24161

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5232  ax-sep 5251  ax-nul 5263  ax-pow 5330  ax-pr 5398  ax-un 7737

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-ov 7417  df-oprab 7418  df-mpo 7419  df-fbas 21583  df-top 23120  df-topon 23137  df-nei 23324  df-fil 24073  df-flim 24166

This theorem is used by:  fclsfnflim  24254