Theorem lmfval 23176

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 23173	. 2 ⊢ ⇝𝑡 = (𝑗 ∈ Top ↦ {⟨𝑓, 𝑥⟩ ∣ (𝑓 ∈ (∪ 𝑗 ↑pm ℂ) ∧ 𝑥 ∈ ∪ 𝑗 ∧ ∀𝑢 ∈ 𝑗 (𝑥 ∈ 𝑢 → ∃𝑦 ∈ ran ℤ≥(𝑓 ↾ 𝑦):𝑦⟶𝑢))})
2		simpr 484	. . . . . . . 8 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑗 = 𝐽) → 𝑗 = 𝐽)
3	2	unieqd 4876	. . . . . . 7 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑗 = 𝐽) → ∪ 𝑗 = ∪ 𝐽)
4		toponuni 22858	. . . . . . . 8 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = ∪ 𝐽)
5	4	adantr 480	. . . . . . 7 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑗 = 𝐽) → 𝑋 = ∪ 𝐽)
6	3, 5	eqtr4d 2774	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑗 = 𝐽) → ∪ 𝑗 = 𝑋)
7	6	oveq1d 7373	. . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑗 = 𝐽) → (∪ 𝑗 ↑pm ℂ) = (𝑋 ↑pm ℂ))
8	7	eleq2d 2822	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑗 = 𝐽) → (𝑓 ∈ (∪ 𝑗 ↑pm ℂ) ↔ 𝑓 ∈ (𝑋 ↑pm ℂ)))
9	6	eleq2d 2822	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑗 = 𝐽) → (𝑥 ∈ ∪ 𝑗 ↔ 𝑥 ∈ 𝑋))
10	2	raleqdv 3296	. . . 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)))
11	8, 9, 10	3anbi123d 1438	. . 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))))
12	11	opabbidv 5164	. 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))})
13		topontop 22857	. 2 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
14		df-3an 1088	. . . . 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)))
15	14	opabbii 5165	. . . 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))}
16		opabssxp 5716	. . . 4 ⊢ {⟨𝑓, 𝑥⟩ ∣ ((𝑓 ∈ (𝑋 ↑pm ℂ) ∧ 𝑥 ∈ 𝑋) ∧ ∀𝑢 ∈ 𝐽 (𝑥 ∈ 𝑢 → ∃𝑦 ∈ ran ℤ≥(𝑓 ↾ 𝑦):𝑦⟶𝑢))} ⊆ ((𝑋 ↑pm ℂ) × 𝑋)
17	15, 16	eqsstri 3980	. . 3 ⊢ {⟨𝑓, 𝑥⟩ ∣ (𝑓 ∈ (𝑋 ↑pm ℂ) ∧ 𝑥 ∈ 𝑋 ∧ ∀𝑢 ∈ 𝐽 (𝑥 ∈ 𝑢 → ∃𝑦 ∈ ran ℤ≥(𝑓 ↾ 𝑦):𝑦⟶𝑢))} ⊆ ((𝑋 ↑pm ℂ) × 𝑋)
18		ovex 7391	. . . 4 ⊢ (𝑋 ↑pm ℂ) ∈ V
19		toponmax 22870	. . . 4 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 ∈ 𝐽)
20		xpexg 7695	. . . 4 ⊢ (((𝑋 ↑pm ℂ) ∈ V ∧ 𝑋 ∈ 𝐽) → ((𝑋 ↑pm ℂ) × 𝑋) ∈ V)
21	18, 19, 20	sylancr 587	. . 3 ⊢ (𝐽 ∈ (TopOn‘𝑋) → ((𝑋 ↑pm ℂ) × 𝑋) ∈ V)
22		ssexg 5268	. . 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)
23	17, 21, 22	sylancr 587	. 2 ⊢ (𝐽 ∈ (TopOn‘𝑋) → {⟨𝑓, 𝑥⟩ ∣ (𝑓 ∈ (𝑋 ↑pm ℂ) ∧ 𝑥 ∈ 𝑋 ∧ ∀𝑢 ∈ 𝐽 (𝑥 ∈ 𝑢 → ∃𝑦 ∈ ran ℤ≥(𝑓 ↾ 𝑦):𝑦⟶𝑢))} ∈ V)
24	1, 12, 13, 23	fvmptd2 6949	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 395   ∧ w3a 1086   = wceq 1541   ∈ wcel 2113  ∀wral 3051  ∃wrex 3060  Vcvv 3440   ⊆ wss 3901  ∪ cuni 4863  {copab 5160   × cxp 5622  ran crn 5625   ↾ cres 5626  ⟶wf 6488  ‘cfv 6492  (class class class)co 7358   ↑_pm cpm 8764  ℂcc 11024  ℤ_≥cuz 12751  Topctop 22837  TopOnctopon 22854  ⇝_𝑡clm 23170

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 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680

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 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-br 5099  df-opab 5161  df-mpt 5180  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-iota 6448  df-fun 6494  df-fv 6500  df-ov 7361  df-top 22838  df-topon 22855  df-lm 23173

This theorem is referenced by:  lmbr  23202  sslm  23243