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Theorem lmfval 22665
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) → (⇝𝑡𝐽) = {⟨𝑓, 𝑥⟩ ∣ (𝑓 ∈ (𝑋pm ℂ) ∧ 𝑥𝑋 ∧ ∀𝑢𝐽 (𝑥𝑢 → ∃𝑦 ∈ ran ℤ(𝑓𝑦):𝑦𝑢))})
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.
StepHypRef Expression
1 df-lm 22662 . 2 𝑡 = (𝑗 ∈ Top ↦ {⟨𝑓, 𝑥⟩ ∣ (𝑓 ∈ ( 𝑗pm ℂ) ∧ 𝑥 𝑗 ∧ ∀𝑢𝑗 (𝑥𝑢 → ∃𝑦 ∈ ran ℤ(𝑓𝑦):𝑦𝑢))})
2 simpr 485 . . . . . . . 8 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑗 = 𝐽) → 𝑗 = 𝐽)
32unieqd 4915 . . . . . . 7 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑗 = 𝐽) → 𝑗 = 𝐽)
4 toponuni 22345 . . . . . . . 8 (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = 𝐽)
54adantr 481 . . . . . . 7 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑗 = 𝐽) → 𝑋 = 𝐽)
63, 5eqtr4d 2774 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑗 = 𝐽) → 𝑗 = 𝑋)
76oveq1d 7408 . . . . 5 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑗 = 𝐽) → ( 𝑗pm ℂ) = (𝑋pm ℂ))
87eleq2d 2818 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑗 = 𝐽) → (𝑓 ∈ ( 𝑗pm ℂ) ↔ 𝑓 ∈ (𝑋pm ℂ)))
96eleq2d 2818 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑗 = 𝐽) → (𝑥 𝑗𝑥𝑋))
102raleqdv 3324 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑗 = 𝐽) → (∀𝑢𝑗 (𝑥𝑢 → ∃𝑦 ∈ ran ℤ(𝑓𝑦):𝑦𝑢) ↔ ∀𝑢𝐽 (𝑥𝑢 → ∃𝑦 ∈ ran ℤ(𝑓𝑦):𝑦𝑢)))
118, 9, 103anbi123d 1436 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑗 = 𝐽) → ((𝑓 ∈ ( 𝑗pm ℂ) ∧ 𝑥 𝑗 ∧ ∀𝑢𝑗 (𝑥𝑢 → ∃𝑦 ∈ ran ℤ(𝑓𝑦):𝑦𝑢)) ↔ (𝑓 ∈ (𝑋pm ℂ) ∧ 𝑥𝑋 ∧ ∀𝑢𝐽 (𝑥𝑢 → ∃𝑦 ∈ ran ℤ(𝑓𝑦):𝑦𝑢))))
1211opabbidv 5207 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑗 = 𝐽) → {⟨𝑓, 𝑥⟩ ∣ (𝑓 ∈ ( 𝑗pm ℂ) ∧ 𝑥 𝑗 ∧ ∀𝑢𝑗 (𝑥𝑢 → ∃𝑦 ∈ ran ℤ(𝑓𝑦):𝑦𝑢))} = {⟨𝑓, 𝑥⟩ ∣ (𝑓 ∈ (𝑋pm ℂ) ∧ 𝑥𝑋 ∧ ∀𝑢𝐽 (𝑥𝑢 → ∃𝑦 ∈ ran ℤ(𝑓𝑦):𝑦𝑢))})
13 topontop 22344 . 2 (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
14 df-3an 1089 . . . . 5 ((𝑓 ∈ (𝑋pm ℂ) ∧ 𝑥𝑋 ∧ ∀𝑢𝐽 (𝑥𝑢 → ∃𝑦 ∈ ran ℤ(𝑓𝑦):𝑦𝑢)) ↔ ((𝑓 ∈ (𝑋pm ℂ) ∧ 𝑥𝑋) ∧ ∀𝑢𝐽 (𝑥𝑢 → ∃𝑦 ∈ ran ℤ(𝑓𝑦):𝑦𝑢)))
1514opabbii 5208 . . . 4 {⟨𝑓, 𝑥⟩ ∣ (𝑓 ∈ (𝑋pm ℂ) ∧ 𝑥𝑋 ∧ ∀𝑢𝐽 (𝑥𝑢 → ∃𝑦 ∈ ran ℤ(𝑓𝑦):𝑦𝑢))} = {⟨𝑓, 𝑥⟩ ∣ ((𝑓 ∈ (𝑋pm ℂ) ∧ 𝑥𝑋) ∧ ∀𝑢𝐽 (𝑥𝑢 → ∃𝑦 ∈ ran ℤ(𝑓𝑦):𝑦𝑢))}
16 opabssxp 5760 . . . 4 {⟨𝑓, 𝑥⟩ ∣ ((𝑓 ∈ (𝑋pm ℂ) ∧ 𝑥𝑋) ∧ ∀𝑢𝐽 (𝑥𝑢 → ∃𝑦 ∈ ran ℤ(𝑓𝑦):𝑦𝑢))} ⊆ ((𝑋pm ℂ) × 𝑋)
1715, 16eqsstri 4012 . . 3 {⟨𝑓, 𝑥⟩ ∣ (𝑓 ∈ (𝑋pm ℂ) ∧ 𝑥𝑋 ∧ ∀𝑢𝐽 (𝑥𝑢 → ∃𝑦 ∈ ran ℤ(𝑓𝑦):𝑦𝑢))} ⊆ ((𝑋pm ℂ) × 𝑋)
18 ovex 7426 . . . 4 (𝑋pm ℂ) ∈ V
19 toponmax 22357 . . . 4 (𝐽 ∈ (TopOn‘𝑋) → 𝑋𝐽)
20 xpexg 7720 . . . 4 (((𝑋pm ℂ) ∈ V ∧ 𝑋𝐽) → ((𝑋pm ℂ) × 𝑋) ∈ V)
2118, 19, 20sylancr 587 . . 3 (𝐽 ∈ (TopOn‘𝑋) → ((𝑋pm ℂ) × 𝑋) ∈ V)
22 ssexg 5316 . . 3 (({⟨𝑓, 𝑥⟩ ∣ (𝑓 ∈ (𝑋pm ℂ) ∧ 𝑥𝑋 ∧ ∀𝑢𝐽 (𝑥𝑢 → ∃𝑦 ∈ ran ℤ(𝑓𝑦):𝑦𝑢))} ⊆ ((𝑋pm ℂ) × 𝑋) ∧ ((𝑋pm ℂ) × 𝑋) ∈ V) → {⟨𝑓, 𝑥⟩ ∣ (𝑓 ∈ (𝑋pm ℂ) ∧ 𝑥𝑋 ∧ ∀𝑢𝐽 (𝑥𝑢 → ∃𝑦 ∈ ran ℤ(𝑓𝑦):𝑦𝑢))} ∈ V)
2317, 21, 22sylancr 587 . 2 (𝐽 ∈ (TopOn‘𝑋) → {⟨𝑓, 𝑥⟩ ∣ (𝑓 ∈ (𝑋pm ℂ) ∧ 𝑥𝑋 ∧ ∀𝑢𝐽 (𝑥𝑢 → ∃𝑦 ∈ ran ℤ(𝑓𝑦):𝑦𝑢))} ∈ V)
241, 12, 13, 23fvmptd2 6992 1 (𝐽 ∈ (TopOn‘𝑋) → (⇝𝑡𝐽) = {⟨𝑓, 𝑥⟩ ∣ (𝑓 ∈ (𝑋pm ℂ) ∧ 𝑥𝑋 ∧ ∀𝑢𝐽 (𝑥𝑢 → ∃𝑦 ∈ ran ℤ(𝑓𝑦):𝑦𝑢))})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1087   = wceq 1541  wcel 2106  wral 3060  wrex 3069  Vcvv 3473  wss 3944   cuni 4901  {copab 5203   × cxp 5667  ran crn 5670  cres 5671  wf 6528  cfv 6532  (class class class)co 7393  pm cpm 8804  cc 11090  cuz 12804  Topctop 22324  TopOnctopon 22341  𝑡clm 22659
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7708
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4523  df-pw 4598  df-sn 4623  df-pr 4625  df-op 4629  df-uni 4902  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-iota 6484  df-fun 6534  df-fv 6540  df-ov 7396  df-top 22325  df-topon 22342  df-lm 22662
This theorem is referenced by:  lmbr  22691  sslm  22732
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