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

Theorem rnmptbddlem 41823
Description: Boundness of the range of a function in maps-to notation. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
Hypotheses
Ref Expression
rnmptbddlem.x 𝑥𝜑
rnmptbddlem.b (𝜑 → ∃𝑦 ∈ ℝ ∀𝑥𝐴 𝐵𝑦)
Assertion
Ref Expression
rnmptbddlem (𝜑 → ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝑦)
Distinct variable groups:   𝑧,𝐴   𝑧,𝐵   𝜑,𝑦,𝑧   𝑥,𝑦,𝑧
Allowed substitution hints:   𝜑(𝑥)   𝐴(𝑥,𝑦)   𝐵(𝑥,𝑦)

Proof of Theorem rnmptbddlem
StepHypRef Expression
1 eqid 2822 . . . . . 6 (𝑥𝐴𝐵) = (𝑥𝐴𝐵)
21elrnmpt 5805 . . . . 5 (𝑧 ∈ V → (𝑧 ∈ ran (𝑥𝐴𝐵) ↔ ∃𝑥𝐴 𝑧 = 𝐵))
32elv 3474 . . . 4 (𝑧 ∈ ran (𝑥𝐴𝐵) ↔ ∃𝑥𝐴 𝑧 = 𝐵)
4 rnmptbddlem.x . . . . . . . 8 𝑥𝜑
5 nfv 1915 . . . . . . . 8 𝑥 𝑦 ∈ ℝ
64, 5nfan 1900 . . . . . . 7 𝑥(𝜑𝑦 ∈ ℝ)
7 nfra1 3208 . . . . . . 7 𝑥𝑥𝐴 𝐵𝑦
86, 7nfan 1900 . . . . . 6 𝑥((𝜑𝑦 ∈ ℝ) ∧ ∀𝑥𝐴 𝐵𝑦)
9 nfv 1915 . . . . . 6 𝑥 𝑧𝑦
10 simp3 1135 . . . . . . . . 9 ((∀𝑥𝐴 𝐵𝑦𝑥𝐴𝑧 = 𝐵) → 𝑧 = 𝐵)
11 rspa 3196 . . . . . . . . . 10 ((∀𝑥𝐴 𝐵𝑦𝑥𝐴) → 𝐵𝑦)
12113adant3 1129 . . . . . . . . 9 ((∀𝑥𝐴 𝐵𝑦𝑥𝐴𝑧 = 𝐵) → 𝐵𝑦)
1310, 12eqbrtrd 5064 . . . . . . . 8 ((∀𝑥𝐴 𝐵𝑦𝑥𝐴𝑧 = 𝐵) → 𝑧𝑦)
14133exp 1116 . . . . . . 7 (∀𝑥𝐴 𝐵𝑦 → (𝑥𝐴 → (𝑧 = 𝐵𝑧𝑦)))
1514adantl 485 . . . . . 6 (((𝜑𝑦 ∈ ℝ) ∧ ∀𝑥𝐴 𝐵𝑦) → (𝑥𝐴 → (𝑧 = 𝐵𝑧𝑦)))
168, 9, 15rexlimd 3303 . . . . 5 (((𝜑𝑦 ∈ ℝ) ∧ ∀𝑥𝐴 𝐵𝑦) → (∃𝑥𝐴 𝑧 = 𝐵𝑧𝑦))
1716imp 410 . . . 4 ((((𝜑𝑦 ∈ ℝ) ∧ ∀𝑥𝐴 𝐵𝑦) ∧ ∃𝑥𝐴 𝑧 = 𝐵) → 𝑧𝑦)
183, 17sylan2b 596 . . 3 ((((𝜑𝑦 ∈ ℝ) ∧ ∀𝑥𝐴 𝐵𝑦) ∧ 𝑧 ∈ ran (𝑥𝐴𝐵)) → 𝑧𝑦)
1918ralrimiva 3174 . 2 (((𝜑𝑦 ∈ ℝ) ∧ ∀𝑥𝐴 𝐵𝑦) → ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝑦)
20 rnmptbddlem.b . 2 (𝜑 → ∃𝑦 ∈ ℝ ∀𝑥𝐴 𝐵𝑦)
2119, 20reximddv3 41729 1 (𝜑 → ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝑦)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  w3a 1084   = wceq 1538  wnf 1785  wcel 2114  wral 3130  wrex 3131  Vcvv 3469   class class class wbr 5042  cmpt 5122  ran crn 5533  cr 10525  cle 10665
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-sep 5179  ax-nul 5186  ax-pr 5307
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ral 3135  df-rex 3136  df-v 3471  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-sn 4540  df-pr 4542  df-op 4546  df-br 5043  df-opab 5105  df-mpt 5123  df-cnv 5540  df-dm 5542  df-rn 5543
This theorem is referenced by:  rnmptbdd  41824
  Copyright terms: Public domain W3C validator