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Theorem lmbr 12457
 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 12434. (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 12436 . . . 4 (𝐽 ∈ (TopOn‘𝑋) → (⇝𝑡𝐽) = {⟨𝑓, 𝑥⟩ ∣ (𝑓 ∈ (𝑋pm ℂ) ∧ 𝑥𝑋 ∧ ∀𝑢𝐽 (𝑥𝑢 → ∃𝑦 ∈ ran ℤ(𝑓𝑦):𝑦𝑢))})
31, 2syl 14 . . 3 (𝜑 → (⇝𝑡𝐽) = {⟨𝑓, 𝑥⟩ ∣ (𝑓 ∈ (𝑋pm ℂ) ∧ 𝑥𝑋 ∧ ∀𝑢𝐽 (𝑥𝑢 → ∃𝑦 ∈ ran ℤ(𝑓𝑦):𝑦𝑢))})
43breqd 3950 . 2 (𝜑 → (𝐹(⇝𝑡𝐽)𝑃𝐹{⟨𝑓, 𝑥⟩ ∣ (𝑓 ∈ (𝑋pm ℂ) ∧ 𝑥𝑋 ∧ ∀𝑢𝐽 (𝑥𝑢 → ∃𝑦 ∈ ran ℤ(𝑓𝑦):𝑦𝑢))}𝑃))
5 reseq1 4825 . . . . . . . . 9 (𝑓 = 𝐹 → (𝑓𝑦) = (𝐹𝑦))
65feq1d 5271 . . . . . . . 8 (𝑓 = 𝐹 → ((𝑓𝑦):𝑦𝑢 ↔ (𝐹𝑦):𝑦𝑢))
76rexbidv 2441 . . . . . . 7 (𝑓 = 𝐹 → (∃𝑦 ∈ ran ℤ(𝑓𝑦):𝑦𝑢 ↔ ∃𝑦 ∈ ran ℤ(𝐹𝑦):𝑦𝑢))
87imbi2d 229 . . . . . 6 (𝑓 = 𝐹 → ((𝑥𝑢 → ∃𝑦 ∈ ran ℤ(𝑓𝑦):𝑦𝑢) ↔ (𝑥𝑢 → ∃𝑦 ∈ ran ℤ(𝐹𝑦):𝑦𝑢)))
98ralbidv 2440 . . . . 5 (𝑓 = 𝐹 → (∀𝑢𝐽 (𝑥𝑢 → ∃𝑦 ∈ ran ℤ(𝑓𝑦):𝑦𝑢) ↔ ∀𝑢𝐽 (𝑥𝑢 → ∃𝑦 ∈ ran ℤ(𝐹𝑦):𝑦𝑢)))
10 eleq1 2204 . . . . . . 7 (𝑥 = 𝑃 → (𝑥𝑢𝑃𝑢))
1110imbi1d 230 . . . . . 6 (𝑥 = 𝑃 → ((𝑥𝑢 → ∃𝑦 ∈ ran ℤ(𝐹𝑦):𝑦𝑢) ↔ (𝑃𝑢 → ∃𝑦 ∈ ran ℤ(𝐹𝑦):𝑦𝑢)))
1211ralbidv 2440 . . . . 5 (𝑥 = 𝑃 → (∀𝑢𝐽 (𝑥𝑢 → ∃𝑦 ∈ ran ℤ(𝐹𝑦):𝑦𝑢) ↔ ∀𝑢𝐽 (𝑃𝑢 → ∃𝑦 ∈ ran ℤ(𝐹𝑦):𝑦𝑢)))
139, 12sylan9bb 458 . . . 4 ((𝑓 = 𝐹𝑥 = 𝑃) → (∀𝑢𝐽 (𝑥𝑢 → ∃𝑦 ∈ ran ℤ(𝑓𝑦):𝑦𝑢) ↔ ∀𝑢𝐽 (𝑃𝑢 → ∃𝑦 ∈ ran ℤ(𝐹𝑦):𝑦𝑢)))
14 df-3an 965 . . . . 5 ((𝑓 ∈ (𝑋pm ℂ) ∧ 𝑥𝑋 ∧ ∀𝑢𝐽 (𝑥𝑢 → ∃𝑦 ∈ ran ℤ(𝑓𝑦):𝑦𝑢)) ↔ ((𝑓 ∈ (𝑋pm ℂ) ∧ 𝑥𝑋) ∧ ∀𝑢𝐽 (𝑥𝑢 → ∃𝑦 ∈ ran ℤ(𝑓𝑦):𝑦𝑢)))
1514opabbii 4005 . . . 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 4604 . . 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 965 . . 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 186 . 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 195 1 (𝜑 → (𝐹(⇝𝑡𝐽)𝑃 ↔ (𝐹 ∈ (𝑋pm ℂ) ∧ 𝑃𝑋 ∧ ∀𝑢𝐽 (𝑃𝑢 → ∃𝑦 ∈ ran ℤ(𝐹𝑦):𝑦𝑢))))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   ∧ w3a 963   = wceq 1332   ∈ wcel 2112  ∀wral 2418  ∃wrex 2419   class class class wbr 3939  {copab 3998  ran crn 4552   ↾ cres 4553  ⟶wf 5131  ‘cfv 5135  (class class class)co 5786   ↑pm cpm 6555  ℂcc 7671  ℤ≥cuz 9379  TopOnctopon 12252  ⇝𝑡clm 12431 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1481  ax-10 1482  ax-11 1483  ax-i12 1484  ax-bndl 1486  ax-4 1487  ax-17 1503  ax-i9 1507  ax-ial 1511  ax-i5r 1512  ax-13 2114  ax-14 2115  ax-ext 2123  ax-sep 4056  ax-pow 4108  ax-pr 4142  ax-un 4366  ax-cnex 7764 This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1732  df-eu 1993  df-mo 1994  df-clab 2128  df-cleq 2134  df-clel 2137  df-nfc 2272  df-ral 2423  df-rex 2424  df-rab 2427  df-v 2693  df-sbc 2916  df-csb 3010  df-un 3082  df-in 3084  df-ss 3091  df-pw 3519  df-sn 3540  df-pr 3541  df-op 3543  df-uni 3747  df-iun 3825  df-br 3940  df-opab 4000  df-mpt 4001  df-id 4226  df-xp 4557  df-rel 4558  df-cnv 4559  df-co 4560  df-dm 4561  df-rn 4562  df-res 4563  df-ima 4564  df-iota 5100  df-fun 5137  df-fn 5138  df-f 5139  df-fv 5143  df-ov 5789  df-oprab 5790  df-mpo 5791  df-1st 6050  df-2nd 6051  df-pm 6557  df-top 12240  df-topon 12253  df-lm 12434 This theorem is referenced by:  lmbr2  12458  lmfpm  12487  lmcl  12489  lmff  12493
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