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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  Ⅎwnf 1785   ∈ wcel 2114  ∀wral 3130  ∃wrex 3131  Vcvv 3469   class class class wbr 5042   ↦ cmpt 5122  ran crn 5533  ℝcr 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
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