Theorem lmfval 14875

Description: The relation "sequence 𝑓 converges to point 𝑦 " in a metric space. (Contributed by NM, 7-Sep-2006.) (Revised by Mario Carneiro, 21-Aug-2015.)

Assertion

Ref	Expression
lmfval	⊢ (𝐽 ∈ (TopOn‘𝑋) → (⇝𝑡‘𝐽) = {⟨𝑓, 𝑥⟩ ∣ (𝑓 ∈ (𝑋 ↑pm ℂ) ∧ 𝑥 ∈ 𝑋 ∧ ∀𝑢 ∈ 𝐽 (𝑥 ∈ 𝑢 → ∃𝑦 ∈ ran ℤ≥(𝑓 ↾ 𝑦):𝑦⟶𝑢))})

Distinct variable groups:   𝑥,𝑓,𝑦,𝑋   𝑢,𝑓,𝐽,𝑥,𝑦

Allowed substitution hint:   𝑋(𝑢)

Proof of Theorem lmfval

Dummy variable 𝑗 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-lm 14872	. . 3 ⊢ ⇝𝑡 = (𝑗 ∈ Top ↦ {⟨𝑓, 𝑥⟩ ∣ (𝑓 ∈ (∪ 𝑗 ↑pm ℂ) ∧ 𝑥 ∈ ∪ 𝑗 ∧ ∀𝑢 ∈ 𝑗 (𝑥 ∈ 𝑢 → ∃𝑦 ∈ ran ℤ≥(𝑓 ↾ 𝑦):𝑦⟶𝑢))})
2	1	a1i 9	. 2 ⊢ (𝐽 ∈ (TopOn‘𝑋) → ⇝𝑡 = (𝑗 ∈ Top ↦ {⟨𝑓, 𝑥⟩ ∣ (𝑓 ∈ (∪ 𝑗 ↑pm ℂ) ∧ 𝑥 ∈ ∪ 𝑗 ∧ ∀𝑢 ∈ 𝑗 (𝑥 ∈ 𝑢 → ∃𝑦 ∈ ran ℤ≥(𝑓 ↾ 𝑦):𝑦⟶𝑢))}))
3		simpr 110	. . . . . . . 8 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑗 = 𝐽) → 𝑗 = 𝐽)
4	3	unieqd 3899	. . . . . . 7 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑗 = 𝐽) → ∪ 𝑗 = ∪ 𝐽)
5		toponuni 14697	. . . . . . . 8 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = ∪ 𝐽)
6	5	adantr 276	. . . . . . 7 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑗 = 𝐽) → 𝑋 = ∪ 𝐽)
7	4, 6	eqtr4d 2265	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑗 = 𝐽) → ∪ 𝑗 = 𝑋)
8	7	oveq1d 6022	. . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑗 = 𝐽) → (∪ 𝑗 ↑pm ℂ) = (𝑋 ↑pm ℂ))
9	8	eleq2d 2299	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑗 = 𝐽) → (𝑓 ∈ (∪ 𝑗 ↑pm ℂ) ↔ 𝑓 ∈ (𝑋 ↑pm ℂ)))
10	7	eleq2d 2299	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑗 = 𝐽) → (𝑥 ∈ ∪ 𝑗 ↔ 𝑥 ∈ 𝑋))
11	3	raleqdv 2734	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑗 = 𝐽) → (∀𝑢 ∈ 𝑗 (𝑥 ∈ 𝑢 → ∃𝑦 ∈ ran ℤ≥(𝑓 ↾ 𝑦):𝑦⟶𝑢) ↔ ∀𝑢 ∈ 𝐽 (𝑥 ∈ 𝑢 → ∃𝑦 ∈ ran ℤ≥(𝑓 ↾ 𝑦):𝑦⟶𝑢)))
12	9, 10, 11	3anbi123d 1346	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑗 = 𝐽) → ((𝑓 ∈ (∪ 𝑗 ↑pm ℂ) ∧ 𝑥 ∈ ∪ 𝑗 ∧ ∀𝑢 ∈ 𝑗 (𝑥 ∈ 𝑢 → ∃𝑦 ∈ ran ℤ≥(𝑓 ↾ 𝑦):𝑦⟶𝑢)) ↔ (𝑓 ∈ (𝑋 ↑pm ℂ) ∧ 𝑥 ∈ 𝑋 ∧ ∀𝑢 ∈ 𝐽 (𝑥 ∈ 𝑢 → ∃𝑦 ∈ ran ℤ≥(𝑓 ↾ 𝑦):𝑦⟶𝑢))))
13	12	opabbidv 4150	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑗 = 𝐽) → {⟨𝑓, 𝑥⟩ ∣ (𝑓 ∈ (∪ 𝑗 ↑pm ℂ) ∧ 𝑥 ∈ ∪ 𝑗 ∧ ∀𝑢 ∈ 𝑗 (𝑥 ∈ 𝑢 → ∃𝑦 ∈ ran ℤ≥(𝑓 ↾ 𝑦):𝑦⟶𝑢))} = {⟨𝑓, 𝑥⟩ ∣ (𝑓 ∈ (𝑋 ↑pm ℂ) ∧ 𝑥 ∈ 𝑋 ∧ ∀𝑢 ∈ 𝐽 (𝑥 ∈ 𝑢 → ∃𝑦 ∈ ran ℤ≥(𝑓 ↾ 𝑦):𝑦⟶𝑢))})
14		topontop 14696	. 2 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
15		df-3an 1004	. . . . 5 ⊢ ((𝑓 ∈ (𝑋 ↑pm ℂ) ∧ 𝑥 ∈ 𝑋 ∧ ∀𝑢 ∈ 𝐽 (𝑥 ∈ 𝑢 → ∃𝑦 ∈ ran ℤ≥(𝑓 ↾ 𝑦):𝑦⟶𝑢)) ↔ ((𝑓 ∈ (𝑋 ↑pm ℂ) ∧ 𝑥 ∈ 𝑋) ∧ ∀𝑢 ∈ 𝐽 (𝑥 ∈ 𝑢 → ∃𝑦 ∈ ran ℤ≥(𝑓 ↾ 𝑦):𝑦⟶𝑢)))
16	15	opabbii 4151	. . . 4 ⊢ {⟨𝑓, 𝑥⟩ ∣ (𝑓 ∈ (𝑋 ↑pm ℂ) ∧ 𝑥 ∈ 𝑋 ∧ ∀𝑢 ∈ 𝐽 (𝑥 ∈ 𝑢 → ∃𝑦 ∈ ran ℤ≥(𝑓 ↾ 𝑦):𝑦⟶𝑢))} = {⟨𝑓, 𝑥⟩ ∣ ((𝑓 ∈ (𝑋 ↑pm ℂ) ∧ 𝑥 ∈ 𝑋) ∧ ∀𝑢 ∈ 𝐽 (𝑥 ∈ 𝑢 → ∃𝑦 ∈ ran ℤ≥(𝑓 ↾ 𝑦):𝑦⟶𝑢))}
17		opabssxp 4793	. . . 4 ⊢ {⟨𝑓, 𝑥⟩ ∣ ((𝑓 ∈ (𝑋 ↑pm ℂ) ∧ 𝑥 ∈ 𝑋) ∧ ∀𝑢 ∈ 𝐽 (𝑥 ∈ 𝑢 → ∃𝑦 ∈ ran ℤ≥(𝑓 ↾ 𝑦):𝑦⟶𝑢))} ⊆ ((𝑋 ↑pm ℂ) × 𝑋)
18	16, 17	eqsstri 3256	. . 3 ⊢ {⟨𝑓, 𝑥⟩ ∣ (𝑓 ∈ (𝑋 ↑pm ℂ) ∧ 𝑥 ∈ 𝑋 ∧ ∀𝑢 ∈ 𝐽 (𝑥 ∈ 𝑢 → ∃𝑦 ∈ ran ℤ≥(𝑓 ↾ 𝑦):𝑦⟶𝑢))} ⊆ ((𝑋 ↑pm ℂ) × 𝑋)
19		fnpm 6811	. . . . 5 ⊢ ↑pm Fn (V × V)
20		toponmax 14707	. . . . . 6 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 ∈ 𝐽)
21	20	elexd 2813	. . . . 5 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 ∈ V)
22		cnex 8131	. . . . . 6 ⊢ ℂ ∈ V
23	22	a1i 9	. . . . 5 ⊢ (𝐽 ∈ (TopOn‘𝑋) → ℂ ∈ V)
24		fnovex 6040	. . . . 5 ⊢ (( ↑pm Fn (V × V) ∧ 𝑋 ∈ V ∧ ℂ ∈ V) → (𝑋 ↑pm ℂ) ∈ V)
25	19, 21, 23, 24	mp3an2i 1376	. . . 4 ⊢ (𝐽 ∈ (TopOn‘𝑋) → (𝑋 ↑pm ℂ) ∈ V)
26		xpexg 4833	. . . 4 ⊢ (((𝑋 ↑pm ℂ) ∈ V ∧ 𝑋 ∈ 𝐽) → ((𝑋 ↑pm ℂ) × 𝑋) ∈ V)
27	25, 20, 26	syl2anc 411	. . 3 ⊢ (𝐽 ∈ (TopOn‘𝑋) → ((𝑋 ↑pm ℂ) × 𝑋) ∈ V)
28		ssexg 4223	. . 3 ⊢ (({⟨𝑓, 𝑥⟩ ∣ (𝑓 ∈ (𝑋 ↑pm ℂ) ∧ 𝑥 ∈ 𝑋 ∧ ∀𝑢 ∈ 𝐽 (𝑥 ∈ 𝑢 → ∃𝑦 ∈ ran ℤ≥(𝑓 ↾ 𝑦):𝑦⟶𝑢))} ⊆ ((𝑋 ↑pm ℂ) × 𝑋) ∧ ((𝑋 ↑pm ℂ) × 𝑋) ∈ V) → {⟨𝑓, 𝑥⟩ ∣ (𝑓 ∈ (𝑋 ↑pm ℂ) ∧ 𝑥 ∈ 𝑋 ∧ ∀𝑢 ∈ 𝐽 (𝑥 ∈ 𝑢 → ∃𝑦 ∈ ran ℤ≥(𝑓 ↾ 𝑦):𝑦⟶𝑢))} ∈ V)
29	18, 27, 28	sylancr 414	. 2 ⊢ (𝐽 ∈ (TopOn‘𝑋) → {⟨𝑓, 𝑥⟩ ∣ (𝑓 ∈ (𝑋 ↑pm ℂ) ∧ 𝑥 ∈ 𝑋 ∧ ∀𝑢 ∈ 𝐽 (𝑥 ∈ 𝑢 → ∃𝑦 ∈ ran ℤ≥(𝑓 ↾ 𝑦):𝑦⟶𝑢))} ∈ V)
30	2, 13, 14, 29	fvmptd 5717	1 ⊢ (𝐽 ∈ (TopOn‘𝑋) → (⇝𝑡‘𝐽) = {⟨𝑓, 𝑥⟩ ∣ (𝑓 ∈ (𝑋 ↑pm ℂ) ∧ 𝑥 ∈ 𝑋 ∧ ∀𝑢 ∈ 𝐽 (𝑥 ∈ 𝑢 → ∃𝑦 ∈ ran ℤ≥(𝑓 ↾ 𝑦):𝑦⟶𝑢))})

Syntax hints:   → wi 4   ∧ wa 104   ∧ w3a 1002   = wceq 1395   ∈ wcel 2200  ∀wral 2508  ∃wrex 2509  Vcvv 2799   ⊆ wss 3197  ∪ cuni 3888  {copab 4144   ↦ cmpt 4145   × cxp 4717  ran crn 4720   ↾ cres 4721   Fn wfn 5313  ⟶wf 5314  ‘cfv 5318  (class class class)co 6007   ↑_pm cpm 6804  ℂcc 8005  ℤ_≥cuz 9730  Topctop 14679  TopOnctopon 14692  ⇝_𝑡clm 14869

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-cnex 8098

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-id 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-fv 5326  df-ov 6010  df-oprab 6011  df-mpo 6012  df-1st 6292  df-2nd 6293  df-pm 6806  df-top 14680  df-topon 14693  df-lm 14872

This theorem is referenced by:  lmbr  14895  sslm  14929