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Theorem lmfval 23256
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 23253 . 2 𝑡 = (𝑗 ∈ Top ↦ {⟨𝑓, 𝑥⟩ ∣ (𝑓 ∈ ( 𝑗pm ℂ) ∧ 𝑥 𝑗 ∧ ∀𝑢𝑗 (𝑥𝑢 → ∃𝑦 ∈ ran ℤ(𝑓𝑦):𝑦𝑢))})
2 simpr 484 . . . . . . . 8 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑗 = 𝐽) → 𝑗 = 𝐽)
32unieqd 4925 . . . . . . 7 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑗 = 𝐽) → 𝑗 = 𝐽)
4 toponuni 22936 . . . . . . . 8 (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = 𝐽)
54adantr 480 . . . . . . 7 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑗 = 𝐽) → 𝑋 = 𝐽)
63, 5eqtr4d 2778 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑗 = 𝐽) → 𝑗 = 𝑋)
76oveq1d 7446 . . . . 5 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑗 = 𝐽) → ( 𝑗pm ℂ) = (𝑋pm ℂ))
87eleq2d 2825 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑗 = 𝐽) → (𝑓 ∈ ( 𝑗pm ℂ) ↔ 𝑓 ∈ (𝑋pm ℂ)))
96eleq2d 2825 . . . 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 1435 . . 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 5214 . 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 22935 . 2 (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
14 df-3an 1088 . . . . 5 ((𝑓 ∈ (𝑋pm ℂ) ∧ 𝑥𝑋 ∧ ∀𝑢𝐽 (𝑥𝑢 → ∃𝑦 ∈ ran ℤ(𝑓𝑦):𝑦𝑢)) ↔ ((𝑓 ∈ (𝑋pm ℂ) ∧ 𝑥𝑋) ∧ ∀𝑢𝐽 (𝑥𝑢 → ∃𝑦 ∈ ran ℤ(𝑓𝑦):𝑦𝑢)))
1514opabbii 5215 . . . 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 5781 . . . 4 {⟨𝑓, 𝑥⟩ ∣ ((𝑓 ∈ (𝑋pm ℂ) ∧ 𝑥𝑋) ∧ ∀𝑢𝐽 (𝑥𝑢 → ∃𝑦 ∈ ran ℤ(𝑓𝑦):𝑦𝑢))} ⊆ ((𝑋pm ℂ) × 𝑋)
1715, 16eqsstri 4030 . . 3 {⟨𝑓, 𝑥⟩ ∣ (𝑓 ∈ (𝑋pm ℂ) ∧ 𝑥𝑋 ∧ ∀𝑢𝐽 (𝑥𝑢 → ∃𝑦 ∈ ran ℤ(𝑓𝑦):𝑦𝑢))} ⊆ ((𝑋pm ℂ) × 𝑋)
18 ovex 7464 . . . 4 (𝑋pm ℂ) ∈ V
19 toponmax 22948 . . . 4 (𝐽 ∈ (TopOn‘𝑋) → 𝑋𝐽)
20 xpexg 7769 . . . 4 (((𝑋pm ℂ) ∈ V ∧ 𝑋𝐽) → ((𝑋pm ℂ) × 𝑋) ∈ V)
2118, 19, 20sylancr 587 . . 3 (𝐽 ∈ (TopOn‘𝑋) → ((𝑋pm ℂ) × 𝑋) ∈ V)
22 ssexg 5329 . . 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 7024 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 395  w3a 1086   = wceq 1537  wcel 2106  wral 3059  wrex 3068  Vcvv 3478  wss 3963   cuni 4912  {copab 5210   × cxp 5687  ran crn 5690  cres 5691  wf 6559  cfv 6563  (class class class)co 7431  pm cpm 8866  cc 11151  cuz 12876  Topctop 22915  TopOnctopon 22932  𝑡clm 23250
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-iota 6516  df-fun 6565  df-fv 6571  df-ov 7434  df-top 22916  df-topon 22933  df-lm 23253
This theorem is referenced by:  lmbr  23282  sslm  23323
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