Users' Mathboxes Mathbox for Glauco Siliprandi < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  rnmptlb Structured version   Visualization version   GIF version

Theorem rnmptlb 45250
Description: Boundness below of the range of a function in maps-to notation. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
Hypothesis
Ref Expression
rnmptlb.1 (𝜑 → ∃𝑦 ∈ ℝ ∀𝑥𝐴 𝑦𝐵)
Assertion
Ref Expression
rnmptlb (𝜑 → ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑦𝑧)
Distinct variable groups:   𝑦,𝐴,𝑧   𝑦,𝐵,𝑧   𝜑,𝑧   𝑥,𝑦,𝑧
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐴(𝑥)   𝐵(𝑥)

Proof of Theorem rnmptlb
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 eqid 2737 . . . . . . 7 (𝑥𝐴𝐵) = (𝑥𝐴𝐵)
21elrnmpt 5969 . . . . . 6 (𝑧 ∈ V → (𝑧 ∈ ran (𝑥𝐴𝐵) ↔ ∃𝑥𝐴 𝑧 = 𝐵))
32elv 3485 . . . . 5 (𝑧 ∈ ran (𝑥𝐴𝐵) ↔ ∃𝑥𝐴 𝑧 = 𝐵)
4 nfra1 3284 . . . . . . . 8 𝑥𝑥𝐴 𝑤𝐵
5 nfv 1914 . . . . . . . 8 𝑥 𝑤𝑧
6 rspa 3248 . . . . . . . . . . 11 ((∀𝑥𝐴 𝑤𝐵𝑥𝐴) → 𝑤𝐵)
763adant3 1133 . . . . . . . . . 10 ((∀𝑥𝐴 𝑤𝐵𝑥𝐴𝑧 = 𝐵) → 𝑤𝐵)
8 simp3 1139 . . . . . . . . . 10 ((∀𝑥𝐴 𝑤𝐵𝑥𝐴𝑧 = 𝐵) → 𝑧 = 𝐵)
97, 8breqtrrd 5171 . . . . . . . . 9 ((∀𝑥𝐴 𝑤𝐵𝑥𝐴𝑧 = 𝐵) → 𝑤𝑧)
1093exp 1120 . . . . . . . 8 (∀𝑥𝐴 𝑤𝐵 → (𝑥𝐴 → (𝑧 = 𝐵𝑤𝑧)))
114, 5, 10rexlimd 3266 . . . . . . 7 (∀𝑥𝐴 𝑤𝐵 → (∃𝑥𝐴 𝑧 = 𝐵𝑤𝑧))
1211imp 406 . . . . . 6 ((∀𝑥𝐴 𝑤𝐵 ∧ ∃𝑥𝐴 𝑧 = 𝐵) → 𝑤𝑧)
1312adantll 714 . . . . 5 ((((𝜑𝑤 ∈ ℝ) ∧ ∀𝑥𝐴 𝑤𝐵) ∧ ∃𝑥𝐴 𝑧 = 𝐵) → 𝑤𝑧)
143, 13sylan2b 594 . . . 4 ((((𝜑𝑤 ∈ ℝ) ∧ ∀𝑥𝐴 𝑤𝐵) ∧ 𝑧 ∈ ran (𝑥𝐴𝐵)) → 𝑤𝑧)
1514ralrimiva 3146 . . 3 (((𝜑𝑤 ∈ ℝ) ∧ ∀𝑥𝐴 𝑤𝐵) → ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑤𝑧)
16 rnmptlb.1 . . . 4 (𝜑 → ∃𝑦 ∈ ℝ ∀𝑥𝐴 𝑦𝐵)
17 breq1 5146 . . . . . 6 (𝑦 = 𝑤 → (𝑦𝐵𝑤𝐵))
1817ralbidv 3178 . . . . 5 (𝑦 = 𝑤 → (∀𝑥𝐴 𝑦𝐵 ↔ ∀𝑥𝐴 𝑤𝐵))
1918cbvrexvw 3238 . . . 4 (∃𝑦 ∈ ℝ ∀𝑥𝐴 𝑦𝐵 ↔ ∃𝑤 ∈ ℝ ∀𝑥𝐴 𝑤𝐵)
2016, 19sylib 218 . . 3 (𝜑 → ∃𝑤 ∈ ℝ ∀𝑥𝐴 𝑤𝐵)
2115, 20reximddv3 3172 . 2 (𝜑 → ∃𝑤 ∈ ℝ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑤𝑧)
22 breq1 5146 . . . 4 (𝑤 = 𝑦 → (𝑤𝑧𝑦𝑧))
2322ralbidv 3178 . . 3 (𝑤 = 𝑦 → (∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑤𝑧 ↔ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑦𝑧))
2423cbvrexvw 3238 . 2 (∃𝑤 ∈ ℝ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑤𝑧 ↔ ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑦𝑧)
2521, 24sylib 218 1 (𝜑 → ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑦𝑧)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1087   = wceq 1540  wcel 2108  wral 3061  wrex 3070  Vcvv 3480   class class class wbr 5143  cmpt 5225  ran crn 5686  cr 11154  cle 11296
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-mpt 5226  df-cnv 5693  df-dm 5695  df-rn 5696
This theorem is referenced by:  infnsuprnmpt  45257  infrpgernmpt  45476
  Copyright terms: Public domain W3C validator