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Theorem rnmptlb 41736
 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 2824 . . . . . . 7 (𝑥𝐴𝐵) = (𝑥𝐴𝐵)
21elrnmpt 5811 . . . . . 6 (𝑧 ∈ V → (𝑧 ∈ ran (𝑥𝐴𝐵) ↔ ∃𝑥𝐴 𝑧 = 𝐵))
32elv 3484 . . . . 5 (𝑧 ∈ ran (𝑥𝐴𝐵) ↔ ∃𝑥𝐴 𝑧 = 𝐵)
4 nfra1 3213 . . . . . . . 8 𝑥𝑥𝐴 𝑤𝐵
5 nfv 1916 . . . . . . . 8 𝑥 𝑤𝑧
6 rspa 3200 . . . . . . . . . . 11 ((∀𝑥𝐴 𝑤𝐵𝑥𝐴) → 𝑤𝐵)
763adant3 1129 . . . . . . . . . 10 ((∀𝑥𝐴 𝑤𝐵𝑥𝐴𝑧 = 𝐵) → 𝑤𝐵)
8 simp3 1135 . . . . . . . . . 10 ((∀𝑥𝐴 𝑤𝐵𝑥𝐴𝑧 = 𝐵) → 𝑧 = 𝐵)
97, 8breqtrrd 5077 . . . . . . . . 9 ((∀𝑥𝐴 𝑤𝐵𝑥𝐴𝑧 = 𝐵) → 𝑤𝑧)
1093exp 1116 . . . . . . . 8 (∀𝑥𝐴 𝑤𝐵 → (𝑥𝐴 → (𝑧 = 𝐵𝑤𝑧)))
114, 5, 10rexlimd 3309 . . . . . . 7 (∀𝑥𝐴 𝑤𝐵 → (∃𝑥𝐴 𝑧 = 𝐵𝑤𝑧))
1211imp 410 . . . . . 6 ((∀𝑥𝐴 𝑤𝐵 ∧ ∃𝑥𝐴 𝑧 = 𝐵) → 𝑤𝑧)
1312adantll 713 . . . . 5 ((((𝜑𝑤 ∈ ℝ) ∧ ∀𝑥𝐴 𝑤𝐵) ∧ ∃𝑥𝐴 𝑧 = 𝐵) → 𝑤𝑧)
143, 13sylan2b 596 . . . 4 ((((𝜑𝑤 ∈ ℝ) ∧ ∀𝑥𝐴 𝑤𝐵) ∧ 𝑧 ∈ ran (𝑥𝐴𝐵)) → 𝑤𝑧)
1514ralrimiva 3176 . . 3 (((𝜑𝑤 ∈ ℝ) ∧ ∀𝑥𝐴 𝑤𝐵) → ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑤𝑧)
16 rnmptlb.1 . . . 4 (𝜑 → ∃𝑦 ∈ ℝ ∀𝑥𝐴 𝑦𝐵)
17 breq1 5052 . . . . . 6 (𝑦 = 𝑤 → (𝑦𝐵𝑤𝐵))
1817ralbidv 3191 . . . . 5 (𝑦 = 𝑤 → (∀𝑥𝐴 𝑦𝐵 ↔ ∀𝑥𝐴 𝑤𝐵))
1918cbvrexvw 3435 . . . 4 (∃𝑦 ∈ ℝ ∀𝑥𝐴 𝑦𝐵 ↔ ∃𝑤 ∈ ℝ ∀𝑥𝐴 𝑤𝐵)
2016, 19sylib 221 . . 3 (𝜑 → ∃𝑤 ∈ ℝ ∀𝑥𝐴 𝑤𝐵)
2115, 20reximddv3 41642 . 2 (𝜑 → ∃𝑤 ∈ ℝ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑤𝑧)
22 breq1 5052 . . . 4 (𝑤 = 𝑦 → (𝑤𝑧𝑦𝑧))
2322ralbidv 3191 . . 3 (𝑤 = 𝑦 → (∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑤𝑧 ↔ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑦𝑧))
2423cbvrexvw 3435 . 2 (∃𝑤 ∈ ℝ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑤𝑧 ↔ ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑦𝑧)
2521, 24sylib 221 1 (𝜑 → ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑦𝑧)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2115  ∀wral 3132  ∃wrex 3133  Vcvv 3479   class class class wbr 5049   ↦ cmpt 5129  ran crn 5539  ℝcr 10523   ≤ cle 10663 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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5186  ax-nul 5193  ax-pr 5313 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 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3137  df-rex 3138  df-rab 3141  df-v 3481  df-dif 3921  df-un 3923  df-in 3925  df-ss 3935  df-nul 4275  df-if 4449  df-sn 4549  df-pr 4551  df-op 4555  df-br 5050  df-opab 5112  df-mpt 5130  df-cnv 5546  df-dm 5548  df-rn 5549 This theorem is referenced by:  infnsuprnmpt  41744  infrpgernmpt  41961
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