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Theorem lmbr 21440
Description: Express the binary relation "sequence 𝐹 converges to point 𝑃 " in a topological space. Definition 1.4-1 of [Kreyszig] p. 25. The condition 𝐹 ⊆ (ℂ × 𝑋) allows us to use objects more general than sequences when convenient; see the comment in df-lm 21411. (Contributed by Mario Carneiro, 14-Nov-2013.)
Hypothesis
Ref Expression
lmbr.2 (𝜑𝐽 ∈ (TopOn‘𝑋))
Assertion
Ref Expression
lmbr (𝜑 → (𝐹(⇝𝑡𝐽)𝑃 ↔ (𝐹 ∈ (𝑋pm ℂ) ∧ 𝑃𝑋 ∧ ∀𝑢𝐽 (𝑃𝑢 → ∃𝑦 ∈ ran ℤ(𝐹𝑦):𝑦𝑢))))
Distinct variable groups:   𝑦,𝑢,𝐹   𝑢,𝐽,𝑦   𝜑,𝑢   𝑢,𝑃   𝑢,𝑋,𝑦
Allowed substitution hints:   𝜑(𝑦)   𝑃(𝑦)

Proof of Theorem lmbr
Dummy variables 𝑓 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 lmbr.2 . . . 4 (𝜑𝐽 ∈ (TopOn‘𝑋))
2 lmfval 21414 . . . 4 (𝐽 ∈ (TopOn‘𝑋) → (⇝𝑡𝐽) = {⟨𝑓, 𝑥⟩ ∣ (𝑓 ∈ (𝑋pm ℂ) ∧ 𝑥𝑋 ∧ ∀𝑢𝐽 (𝑥𝑢 → ∃𝑦 ∈ ran ℤ(𝑓𝑦):𝑦𝑢))})
31, 2syl 17 . . 3 (𝜑 → (⇝𝑡𝐽) = {⟨𝑓, 𝑥⟩ ∣ (𝑓 ∈ (𝑋pm ℂ) ∧ 𝑥𝑋 ∧ ∀𝑢𝐽 (𝑥𝑢 → ∃𝑦 ∈ ran ℤ(𝑓𝑦):𝑦𝑢))})
43breqd 4886 . 2 (𝜑 → (𝐹(⇝𝑡𝐽)𝑃𝐹{⟨𝑓, 𝑥⟩ ∣ (𝑓 ∈ (𝑋pm ℂ) ∧ 𝑥𝑋 ∧ ∀𝑢𝐽 (𝑥𝑢 → ∃𝑦 ∈ ran ℤ(𝑓𝑦):𝑦𝑢))}𝑃))
5 reseq1 5627 . . . . . . . . 9 (𝑓 = 𝐹 → (𝑓𝑦) = (𝐹𝑦))
65feq1d 6267 . . . . . . . 8 (𝑓 = 𝐹 → ((𝑓𝑦):𝑦𝑢 ↔ (𝐹𝑦):𝑦𝑢))
76rexbidv 3262 . . . . . . 7 (𝑓 = 𝐹 → (∃𝑦 ∈ ran ℤ(𝑓𝑦):𝑦𝑢 ↔ ∃𝑦 ∈ ran ℤ(𝐹𝑦):𝑦𝑢))
87imbi2d 332 . . . . . 6 (𝑓 = 𝐹 → ((𝑥𝑢 → ∃𝑦 ∈ ran ℤ(𝑓𝑦):𝑦𝑢) ↔ (𝑥𝑢 → ∃𝑦 ∈ ran ℤ(𝐹𝑦):𝑦𝑢)))
98ralbidv 3195 . . . . 5 (𝑓 = 𝐹 → (∀𝑢𝐽 (𝑥𝑢 → ∃𝑦 ∈ ran ℤ(𝑓𝑦):𝑦𝑢) ↔ ∀𝑢𝐽 (𝑥𝑢 → ∃𝑦 ∈ ran ℤ(𝐹𝑦):𝑦𝑢)))
10 eleq1 2894 . . . . . . 7 (𝑥 = 𝑃 → (𝑥𝑢𝑃𝑢))
1110imbi1d 333 . . . . . 6 (𝑥 = 𝑃 → ((𝑥𝑢 → ∃𝑦 ∈ ran ℤ(𝐹𝑦):𝑦𝑢) ↔ (𝑃𝑢 → ∃𝑦 ∈ ran ℤ(𝐹𝑦):𝑦𝑢)))
1211ralbidv 3195 . . . . 5 (𝑥 = 𝑃 → (∀𝑢𝐽 (𝑥𝑢 → ∃𝑦 ∈ ran ℤ(𝐹𝑦):𝑦𝑢) ↔ ∀𝑢𝐽 (𝑃𝑢 → ∃𝑦 ∈ ran ℤ(𝐹𝑦):𝑦𝑢)))
139, 12sylan9bb 505 . . . 4 ((𝑓 = 𝐹𝑥 = 𝑃) → (∀𝑢𝐽 (𝑥𝑢 → ∃𝑦 ∈ ran ℤ(𝑓𝑦):𝑦𝑢) ↔ ∀𝑢𝐽 (𝑃𝑢 → ∃𝑦 ∈ ran ℤ(𝐹𝑦):𝑦𝑢)))
14 df-3an 1113 . . . . 5 ((𝑓 ∈ (𝑋pm ℂ) ∧ 𝑥𝑋 ∧ ∀𝑢𝐽 (𝑥𝑢 → ∃𝑦 ∈ ran ℤ(𝑓𝑦):𝑦𝑢)) ↔ ((𝑓 ∈ (𝑋pm ℂ) ∧ 𝑥𝑋) ∧ ∀𝑢𝐽 (𝑥𝑢 → ∃𝑦 ∈ ran ℤ(𝑓𝑦):𝑦𝑢)))
1514opabbii 4942 . . . 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))}
1613, 15brab2a 5433 . . 3 (𝐹{⟨𝑓, 𝑥⟩ ∣ (𝑓 ∈ (𝑋pm ℂ) ∧ 𝑥𝑋 ∧ ∀𝑢𝐽 (𝑥𝑢 → ∃𝑦 ∈ ran ℤ(𝑓𝑦):𝑦𝑢))}𝑃 ↔ ((𝐹 ∈ (𝑋pm ℂ) ∧ 𝑃𝑋) ∧ ∀𝑢𝐽 (𝑃𝑢 → ∃𝑦 ∈ ran ℤ(𝐹𝑦):𝑦𝑢)))
17 df-3an 1113 . . 3 ((𝐹 ∈ (𝑋pm ℂ) ∧ 𝑃𝑋 ∧ ∀𝑢𝐽 (𝑃𝑢 → ∃𝑦 ∈ ran ℤ(𝐹𝑦):𝑦𝑢)) ↔ ((𝐹 ∈ (𝑋pm ℂ) ∧ 𝑃𝑋) ∧ ∀𝑢𝐽 (𝑃𝑢 → ∃𝑦 ∈ ran ℤ(𝐹𝑦):𝑦𝑢)))
1816, 17bitr4i 270 . 2 (𝐹{⟨𝑓, 𝑥⟩ ∣ (𝑓 ∈ (𝑋pm ℂ) ∧ 𝑥𝑋 ∧ ∀𝑢𝐽 (𝑥𝑢 → ∃𝑦 ∈ ran ℤ(𝑓𝑦):𝑦𝑢))}𝑃 ↔ (𝐹 ∈ (𝑋pm ℂ) ∧ 𝑃𝑋 ∧ ∀𝑢𝐽 (𝑃𝑢 → ∃𝑦 ∈ ran ℤ(𝐹𝑦):𝑦𝑢)))
194, 18syl6bb 279 1 (𝜑 → (𝐹(⇝𝑡𝐽)𝑃 ↔ (𝐹 ∈ (𝑋pm ℂ) ∧ 𝑃𝑋 ∧ ∀𝑢𝐽 (𝑃𝑢 → ∃𝑦 ∈ ran ℤ(𝐹𝑦):𝑦𝑢))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 386  w3a 1111   = wceq 1656  wcel 2164  wral 3117  wrex 3118   class class class wbr 4875  {copab 4937  ran crn 5347  cres 5348  wf 6123  cfv 6127  (class class class)co 6910  pm cpm 8128  cc 10257  cuz 11975  TopOnctopon 21092  𝑡clm 21408
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-sep 5007  ax-nul 5015  ax-pow 5067  ax-pr 5129  ax-un 7214
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4147  df-if 4309  df-pw 4382  df-sn 4400  df-pr 4402  df-op 4406  df-uni 4661  df-br 4876  df-opab 4938  df-mpt 4955  df-id 5252  df-xp 5352  df-rel 5353  df-cnv 5354  df-co 5355  df-dm 5356  df-rn 5357  df-res 5358  df-iota 6090  df-fun 6129  df-fn 6130  df-f 6131  df-fv 6135  df-ov 6913  df-top 21076  df-topon 21093  df-lm 21411
This theorem is referenced by:  lmbr2  21441  lmfpm  21477  lmcl  21479  lmff  21483  lmmbr  23433
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