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Theorem lmbr 21284
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 21255. (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 21258 . . . 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 4815 . 2 (𝜑 → (𝐹(⇝𝑡𝐽)𝑃𝐹{⟨𝑓, 𝑥⟩ ∣ (𝑓 ∈ (𝑋pm ℂ) ∧ 𝑥𝑋 ∧ ∀𝑢𝐽 (𝑥𝑢 → ∃𝑦 ∈ ran ℤ(𝑓𝑦):𝑦𝑢))}𝑃))
5 reseq1 5545 . . . . . . . . 9 (𝑓 = 𝐹 → (𝑓𝑦) = (𝐹𝑦))
65feq1d 6191 . . . . . . . 8 (𝑓 = 𝐹 → ((𝑓𝑦):𝑦𝑢 ↔ (𝐹𝑦):𝑦𝑢))
76rexbidv 3190 . . . . . . 7 (𝑓 = 𝐹 → (∃𝑦 ∈ ran ℤ(𝑓𝑦):𝑦𝑢 ↔ ∃𝑦 ∈ ran ℤ(𝐹𝑦):𝑦𝑢))
87imbi2d 329 . . . . . 6 (𝑓 = 𝐹 → ((𝑥𝑢 → ∃𝑦 ∈ ran ℤ(𝑓𝑦):𝑦𝑢) ↔ (𝑥𝑢 → ∃𝑦 ∈ ran ℤ(𝐹𝑦):𝑦𝑢)))
98ralbidv 3124 . . . . 5 (𝑓 = 𝐹 → (∀𝑢𝐽 (𝑥𝑢 → ∃𝑦 ∈ ran ℤ(𝑓𝑦):𝑦𝑢) ↔ ∀𝑢𝐽 (𝑥𝑢 → ∃𝑦 ∈ ran ℤ(𝐹𝑦):𝑦𝑢)))
10 eleq1 2827 . . . . . . 7 (𝑥 = 𝑃 → (𝑥𝑢𝑃𝑢))
1110imbi1d 330 . . . . . 6 (𝑥 = 𝑃 → ((𝑥𝑢 → ∃𝑦 ∈ ran ℤ(𝐹𝑦):𝑦𝑢) ↔ (𝑃𝑢 → ∃𝑦 ∈ ran ℤ(𝐹𝑦):𝑦𝑢)))
1211ralbidv 3124 . . . . 5 (𝑥 = 𝑃 → (∀𝑢𝐽 (𝑥𝑢 → ∃𝑦 ∈ ran ℤ(𝐹𝑦):𝑦𝑢) ↔ ∀𝑢𝐽 (𝑃𝑢 → ∃𝑦 ∈ ran ℤ(𝐹𝑦):𝑦𝑢)))
139, 12sylan9bb 738 . . . 4 ((𝑓 = 𝐹𝑥 = 𝑃) → (∀𝑢𝐽 (𝑥𝑢 → ∃𝑦 ∈ ran ℤ(𝑓𝑦):𝑦𝑢) ↔ ∀𝑢𝐽 (𝑃𝑢 → ∃𝑦 ∈ ran ℤ(𝐹𝑦):𝑦𝑢)))
14 df-3an 1074 . . . . 5 ((𝑓 ∈ (𝑋pm ℂ) ∧ 𝑥𝑋 ∧ ∀𝑢𝐽 (𝑥𝑢 → ∃𝑦 ∈ ran ℤ(𝑓𝑦):𝑦𝑢)) ↔ ((𝑓 ∈ (𝑋pm ℂ) ∧ 𝑥𝑋) ∧ ∀𝑢𝐽 (𝑥𝑢 → ∃𝑦 ∈ ran ℤ(𝑓𝑦):𝑦𝑢)))
1514opabbii 4869 . . . 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 5351 . . 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 1074 . . 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 267 . 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 276 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 196  wa 383  w3a 1072   = wceq 1632  wcel 2139  wral 3050  wrex 3051   class class class wbr 4804  {copab 4864  ran crn 5267  cres 5268  wf 6045  cfv 6049  (class class class)co 6814  pm cpm 8026  cc 10146  cuz 11899  TopOnctopon 20937  𝑡clm 21252
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-fv 6057  df-ov 6817  df-top 20921  df-topon 20938  df-lm 21255
This theorem is referenced by:  lmbr2  21285  lmfpm  21321  lmcl  21323  lmff  21327  lmmbr  23276
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