Theorem lmreltop 14750

Description: The topological space convergence relation is a relation. (Contributed by Jim Kingdon, 25-Mar-2023.)

Assertion

Ref	Expression
lmreltop	⊢ (𝐽 ∈ Top → Rel (⇝𝑡‘𝐽))

Proof of Theorem lmreltop

Dummy variables 𝑓 𝑢 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		relopab 4817	. 2 ⊢ Rel {⟨𝑓, 𝑥⟩ ∣ (𝑓 ∈ (∪ 𝐽 ↑pm ℂ) ∧ 𝑥 ∈ ∪ 𝐽 ∧ ∀𝑢 ∈ 𝐽 (𝑥 ∈ 𝑢 → ∃𝑦 ∈ ran ℤ≥(𝑓 ↾ 𝑦):𝑦⟶𝑢))}
2		toptopon2 14576	. . . 4 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘∪ 𝐽))
3		lmfval 14749	. . . 4 ⊢ (𝐽 ∈ (TopOn‘∪ 𝐽) → (⇝𝑡‘𝐽) = {⟨𝑓, 𝑥⟩ ∣ (𝑓 ∈ (∪ 𝐽 ↑pm ℂ) ∧ 𝑥 ∈ ∪ 𝐽 ∧ ∀𝑢 ∈ 𝐽 (𝑥 ∈ 𝑢 → ∃𝑦 ∈ ran ℤ≥(𝑓 ↾ 𝑦):𝑦⟶𝑢))})
4	2, 3	sylbi 121	. . 3 ⊢ (𝐽 ∈ Top → (⇝𝑡‘𝐽) = {⟨𝑓, 𝑥⟩ ∣ (𝑓 ∈ (∪ 𝐽 ↑pm ℂ) ∧ 𝑥 ∈ ∪ 𝐽 ∧ ∀𝑢 ∈ 𝐽 (𝑥 ∈ 𝑢 → ∃𝑦 ∈ ran ℤ≥(𝑓 ↾ 𝑦):𝑦⟶𝑢))})
5	4	releqd 4772	. 2 ⊢ (𝐽 ∈ Top → (Rel (⇝𝑡‘𝐽) ↔ Rel {⟨𝑓, 𝑥⟩ ∣ (𝑓 ∈ (∪ 𝐽 ↑pm ℂ) ∧ 𝑥 ∈ ∪ 𝐽 ∧ ∀𝑢 ∈ 𝐽 (𝑥 ∈ 𝑢 → ∃𝑦 ∈ ran ℤ≥(𝑓 ↾ 𝑦):𝑦⟶𝑢))}))
6	1, 5	mpbiri 168	1 ⊢ (𝐽 ∈ Top → Rel (⇝𝑡‘𝐽))

Syntax hints:   → wi 4   ∧ w3a 981   = wceq 1373   ∈ wcel 2177  ∀wral 2485  ∃wrex 2486  ∪ cuni 3859  {copab 4115  ran crn 4689   ↾ cres 4690  Rel wrel 4693  ⟶wf 5281  ‘cfv 5285  (class class class)co 5962   ↑_pm cpm 6754  ℂcc 7953  ℤ_≥cuz 9678  Topctop 14554  TopOnctopon 14567  ⇝_𝑡clm 14744

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-sep 4173  ax-pow 4229  ax-pr 4264  ax-un 4493  ax-cnex 8046

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ral 2490  df-rex 2491  df-rab 2494  df-v 2775  df-sbc 3003  df-csb 3098  df-un 3174  df-in 3176  df-ss 3183  df-pw 3623  df-sn 3644  df-pr 3645  df-op 3647  df-uni 3860  df-iun 3938  df-br 4055  df-opab 4117  df-mpt 4118  df-id 4353  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-iota 5246  df-fun 5287  df-fn 5288  df-f 5289  df-fv 5293  df-ov 5965  df-oprab 5966  df-mpo 5967  df-1st 6244  df-2nd 6245  df-pm 6756  df-top 14555  df-topon 14568  df-lm 14747

This theorem is referenced by: (None)