Theorem lmfval 14428

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 14426	. . 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 3850	. . . . . . 7 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑗 = 𝐽) → ∪ 𝑗 = ∪ 𝐽)
5		toponuni 14251	. . . . . . . 8 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = ∪ 𝐽)
6	5	adantr 276	. . . . . . 7 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑗 = 𝐽) → 𝑋 = ∪ 𝐽)
7	4, 6	eqtr4d 2232	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑗 = 𝐽) → ∪ 𝑗 = 𝑋)
8	7	oveq1d 5937	. . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑗 = 𝐽) → (∪ 𝑗 ↑pm ℂ) = (𝑋 ↑pm ℂ))
9	8	eleq2d 2266	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑗 = 𝐽) → (𝑓 ∈ (∪ 𝑗 ↑pm ℂ) ↔ 𝑓 ∈ (𝑋 ↑pm ℂ)))
10	7	eleq2d 2266	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑗 = 𝐽) → (𝑥 ∈ ∪ 𝑗 ↔ 𝑥 ∈ 𝑋))
11	3	raleqdv 2699	. . . 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 1323	. . 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 4099	. 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 14250	. 2 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
15		df-3an 982	. . . . 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 4100	. . . 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 4737	. . . 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 3215	. . 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 6715	. . . . 5 ⊢ ↑pm Fn (V × V)
20		toponmax 14261	. . . . . 6 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 ∈ 𝐽)
21	20	elexd 2776	. . . . 5 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 ∈ V)
22		cnex 8003	. . . . . 6 ⊢ ℂ ∈ V
23	22	a1i 9	. . . . 5 ⊢ (𝐽 ∈ (TopOn‘𝑋) → ℂ ∈ V)
24		fnovex 5955	. . . . 5 ⊢ (( ↑pm Fn (V × V) ∧ 𝑋 ∈ V ∧ ℂ ∈ V) → (𝑋 ↑pm ℂ) ∈ V)
25	19, 21, 23, 24	mp3an2i 1353	. . . 4 ⊢ (𝐽 ∈ (TopOn‘𝑋) → (𝑋 ↑pm ℂ) ∈ V)
26		xpexg 4777	. . . 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 4172	. . 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 5642	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 980   = wceq 1364   ∈ wcel 2167  ∀wral 2475  ∃wrex 2476  Vcvv 2763   ⊆ wss 3157  ∪ cuni 3839  {copab 4093   ↦ cmpt 4094   × cxp 4661  ran crn 4664   ↾ cres 4665   Fn wfn 5253  ⟶wf 5254  ‘cfv 5258  (class class class)co 5922   ↑_pm cpm 6708  ℂcc 7877  ℤ_≥cuz 9601  Topctop 14233  TopOnctopon 14246  ⇝_𝑡clm 14423

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-cnex 7970

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-un 3161  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-iun 3918  df-br 4034  df-opab 4095  df-mpt 4096  df-id 4328  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-fv 5266  df-ov 5925  df-oprab 5926  df-mpo 5927  df-1st 6198  df-2nd 6199  df-pm 6710  df-top 14234  df-topon 14247  df-lm 14426

This theorem is referenced by:  lmreltop  14429  lmbr  14449  sslm  14483