Theorem flfnei 22591

Description: The property of being a limit point of a function in terms of neighborhoods. (Contributed by Jeff Hankins, 9-Nov-2009.) (Revised by Stefan O'Rear, 6-Aug-2015.)

Assertion

Ref	Expression
flfnei	⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → (𝐴 ∈ ((𝐽 fLimf 𝐿)‘𝐹) ↔ (𝐴 ∈ 𝑋 ∧ ∀𝑛 ∈ ((nei‘𝐽)‘{𝐴})∃𝑠 ∈ 𝐿 (𝐹 “ 𝑠) ⊆ 𝑛)))

Distinct variable groups:   𝑛,𝑠,𝐹   𝐴,𝑛   𝑛,𝐽,𝑠   𝑛,𝐿,𝑠   𝑛,𝑋,𝑠   𝑛,𝑌,𝑠

Allowed substitution hint:   𝐴(𝑠)

Proof of Theorem flfnei

Step	Hyp	Ref	Expression
1		flfval 22590	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → ((𝐽 fLimf 𝐿)‘𝐹) = (𝐽 fLim ((𝑋 FilMap 𝐹)‘𝐿)))
2	1	eleq2d 2896	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → (𝐴 ∈ ((𝐽 fLimf 𝐿)‘𝐹) ↔ 𝐴 ∈ (𝐽 fLim ((𝑋 FilMap 𝐹)‘𝐿))))
3		simp1 1131	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → 𝐽 ∈ (TopOn‘𝑋))
4		toponmax 21526	. . . . 5 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 ∈ 𝐽)
5	4	3ad2ant1 1128	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → 𝑋 ∈ 𝐽)
6		filfbas 22448	. . . . 5 ⊢ (𝐿 ∈ (Fil‘𝑌) → 𝐿 ∈ (fBas‘𝑌))
7	6	3ad2ant2 1129	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → 𝐿 ∈ (fBas‘𝑌))
8		simp3 1133	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → 𝐹:𝑌⟶𝑋)
9		fmfil 22544	. . . 4 ⊢ ((𝑋 ∈ 𝐽 ∧ 𝐿 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → ((𝑋 FilMap 𝐹)‘𝐿) ∈ (Fil‘𝑋))
10	5, 7, 8, 9	syl3anc 1366	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → ((𝑋 FilMap 𝐹)‘𝐿) ∈ (Fil‘𝑋))
11		elflim 22571	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ ((𝑋 FilMap 𝐹)‘𝐿) ∈ (Fil‘𝑋)) → (𝐴 ∈ (𝐽 fLim ((𝑋 FilMap 𝐹)‘𝐿)) ↔ (𝐴 ∈ 𝑋 ∧ ((nei‘𝐽)‘{𝐴}) ⊆ ((𝑋 FilMap 𝐹)‘𝐿))))
12	3, 10, 11	syl2anc 586	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → (𝐴 ∈ (𝐽 fLim ((𝑋 FilMap 𝐹)‘𝐿)) ↔ (𝐴 ∈ 𝑋 ∧ ((nei‘𝐽)‘{𝐴}) ⊆ ((𝑋 FilMap 𝐹)‘𝐿))))
13		dfss3 3954	. . . 4 ⊢ (((nei‘𝐽)‘{𝐴}) ⊆ ((𝑋 FilMap 𝐹)‘𝐿) ↔ ∀𝑛 ∈ ((nei‘𝐽)‘{𝐴})𝑛 ∈ ((𝑋 FilMap 𝐹)‘𝐿))
14		topontop 21513	. . . . . . . . 9 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
15	14	3ad2ant1 1128	. . . . . . . 8 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → 𝐽 ∈ Top)
16		eqid 2819	. . . . . . . . 9 ⊢ ∪ 𝐽 = ∪ 𝐽
17	16	neii1 21706	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ 𝑛 ∈ ((nei‘𝐽)‘{𝐴})) → 𝑛 ⊆ ∪ 𝐽)
18	15, 17	sylan 582	. . . . . . 7 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ 𝑛 ∈ ((nei‘𝐽)‘{𝐴})) → 𝑛 ⊆ ∪ 𝐽)
19		toponuni 21514	. . . . . . . . 9 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = ∪ 𝐽)
20	19	3ad2ant1 1128	. . . . . . . 8 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → 𝑋 = ∪ 𝐽)
21	20	adantr 483	. . . . . . 7 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ 𝑛 ∈ ((nei‘𝐽)‘{𝐴})) → 𝑋 = ∪ 𝐽)
22	18, 21	sseqtrrd 4006	. . . . . 6 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ 𝑛 ∈ ((nei‘𝐽)‘{𝐴})) → 𝑛 ⊆ 𝑋)
23		elfm 22547	. . . . . . . 8 ⊢ ((𝑋 ∈ 𝐽 ∧ 𝐿 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → (𝑛 ∈ ((𝑋 FilMap 𝐹)‘𝐿) ↔ (𝑛 ⊆ 𝑋 ∧ ∃𝑠 ∈ 𝐿 (𝐹 “ 𝑠) ⊆ 𝑛)))
24	5, 7, 8, 23	syl3anc 1366	. . . . . . 7 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → (𝑛 ∈ ((𝑋 FilMap 𝐹)‘𝐿) ↔ (𝑛 ⊆ 𝑋 ∧ ∃𝑠 ∈ 𝐿 (𝐹 “ 𝑠) ⊆ 𝑛)))
25	24	baibd 542	. . . . . 6 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ 𝑛 ⊆ 𝑋) → (𝑛 ∈ ((𝑋 FilMap 𝐹)‘𝐿) ↔ ∃𝑠 ∈ 𝐿 (𝐹 “ 𝑠) ⊆ 𝑛))
26	22, 25	syldan 593	. . . . 5 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ 𝑛 ∈ ((nei‘𝐽)‘{𝐴})) → (𝑛 ∈ ((𝑋 FilMap 𝐹)‘𝐿) ↔ ∃𝑠 ∈ 𝐿 (𝐹 “ 𝑠) ⊆ 𝑛))
27	26	ralbidva 3194	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → (∀𝑛 ∈ ((nei‘𝐽)‘{𝐴})𝑛 ∈ ((𝑋 FilMap 𝐹)‘𝐿) ↔ ∀𝑛 ∈ ((nei‘𝐽)‘{𝐴})∃𝑠 ∈ 𝐿 (𝐹 “ 𝑠) ⊆ 𝑛))
28	13, 27	syl5bb 285	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → (((nei‘𝐽)‘{𝐴}) ⊆ ((𝑋 FilMap 𝐹)‘𝐿) ↔ ∀𝑛 ∈ ((nei‘𝐽)‘{𝐴})∃𝑠 ∈ 𝐿 (𝐹 “ 𝑠) ⊆ 𝑛))
29	28	anbi2d 630	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → ((𝐴 ∈ 𝑋 ∧ ((nei‘𝐽)‘{𝐴}) ⊆ ((𝑋 FilMap 𝐹)‘𝐿)) ↔ (𝐴 ∈ 𝑋 ∧ ∀𝑛 ∈ ((nei‘𝐽)‘{𝐴})∃𝑠 ∈ 𝐿 (𝐹 “ 𝑠) ⊆ 𝑛)))
30	2, 12, 29	3bitrd 307	1 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → (𝐴 ∈ ((𝐽 fLimf 𝐿)‘𝐹) ↔ (𝐴 ∈ 𝑋 ∧ ∀𝑛 ∈ ((nei‘𝐽)‘{𝐴})∃𝑠 ∈ 𝐿 (𝐹 “ 𝑠) ⊆ 𝑛)))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   ∧ w3a 1082   = wceq 1531   ∈ wcel 2108  ∀wral 3136  ∃wrex 3137   ⊆ wss 3934  {csn 4559  ∪ cuni 4830   “ cima 5551  ⟶wf 6344  ‘cfv 6348  (class class class)co 7148  fBascfbas 20525  Topctop 21493  TopOnctopon 21510  neicnei 21697  Filcfil 22445   FilMap cfm 22533   fLim cflim 22534   fLimf cflf 22535

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1905  ax-6 1964  ax-7 2009  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2154  ax-12 2170  ax-ext 2791  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7453

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1084  df-tru 1534  df-ex 1775  df-nf 1779  df-sb 2064  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-nel 3122  df-ral 3141  df-rex 3142  df-reu 3143  df-rab 3145  df-v 3495  df-sbc 3771  df-csb 3882  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-op 4566  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-ov 7151  df-oprab 7152  df-mpo 7153  df-map 8400  df-fbas 20534  df-fg 20535  df-top 21494  df-topon 21511  df-nei 21698  df-fil 22446  df-fm 22538  df-flim 22539  df-flf 22540

This theorem is referenced by:  flfneii  22592  cnextcn  22667  cnextfres1  22668