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Theorem rnmptlb 42758
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 2738 . . . . . . 7 (𝑥𝐴𝐵) = (𝑥𝐴𝐵)
21elrnmpt 5867 . . . . . 6 (𝑧 ∈ V → (𝑧 ∈ ran (𝑥𝐴𝐵) ↔ ∃𝑥𝐴 𝑧 = 𝐵))
32elv 3437 . . . . 5 (𝑧 ∈ ran (𝑥𝐴𝐵) ↔ ∃𝑥𝐴 𝑧 = 𝐵)
4 nfra1 3144 . . . . . . . 8 𝑥𝑥𝐴 𝑤𝐵
5 nfv 1917 . . . . . . . 8 𝑥 𝑤𝑧
6 rspa 3132 . . . . . . . . . . 11 ((∀𝑥𝐴 𝑤𝐵𝑥𝐴) → 𝑤𝐵)
763adant3 1131 . . . . . . . . . 10 ((∀𝑥𝐴 𝑤𝐵𝑥𝐴𝑧 = 𝐵) → 𝑤𝐵)
8 simp3 1137 . . . . . . . . . 10 ((∀𝑥𝐴 𝑤𝐵𝑥𝐴𝑧 = 𝐵) → 𝑧 = 𝐵)
97, 8breqtrrd 5104 . . . . . . . . 9 ((∀𝑥𝐴 𝑤𝐵𝑥𝐴𝑧 = 𝐵) → 𝑤𝑧)
1093exp 1118 . . . . . . . 8 (∀𝑥𝐴 𝑤𝐵 → (𝑥𝐴 → (𝑧 = 𝐵𝑤𝑧)))
114, 5, 10rexlimd 3249 . . . . . . 7 (∀𝑥𝐴 𝑤𝐵 → (∃𝑥𝐴 𝑧 = 𝐵𝑤𝑧))
1211imp 407 . . . . . 6 ((∀𝑥𝐴 𝑤𝐵 ∧ ∃𝑥𝐴 𝑧 = 𝐵) → 𝑤𝑧)
1312adantll 711 . . . . 5 ((((𝜑𝑤 ∈ ℝ) ∧ ∀𝑥𝐴 𝑤𝐵) ∧ ∃𝑥𝐴 𝑧 = 𝐵) → 𝑤𝑧)
143, 13sylan2b 594 . . . 4 ((((𝜑𝑤 ∈ ℝ) ∧ ∀𝑥𝐴 𝑤𝐵) ∧ 𝑧 ∈ ran (𝑥𝐴𝐵)) → 𝑤𝑧)
1514ralrimiva 3103 . . 3 (((𝜑𝑤 ∈ ℝ) ∧ ∀𝑥𝐴 𝑤𝐵) → ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑤𝑧)
16 rnmptlb.1 . . . 4 (𝜑 → ∃𝑦 ∈ ℝ ∀𝑥𝐴 𝑦𝐵)
17 breq1 5079 . . . . . 6 (𝑦 = 𝑤 → (𝑦𝐵𝑤𝐵))
1817ralbidv 3119 . . . . 5 (𝑦 = 𝑤 → (∀𝑥𝐴 𝑦𝐵 ↔ ∀𝑥𝐴 𝑤𝐵))
1918cbvrexvw 3383 . . . 4 (∃𝑦 ∈ ℝ ∀𝑥𝐴 𝑦𝐵 ↔ ∃𝑤 ∈ ℝ ∀𝑥𝐴 𝑤𝐵)
2016, 19sylib 217 . . 3 (𝜑 → ∃𝑤 ∈ ℝ ∀𝑥𝐴 𝑤𝐵)
2115, 20reximddv3 42670 . 2 (𝜑 → ∃𝑤 ∈ ℝ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑤𝑧)
22 breq1 5079 . . . 4 (𝑤 = 𝑦 → (𝑤𝑧𝑦𝑧))
2322ralbidv 3119 . . 3 (𝑤 = 𝑦 → (∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑤𝑧 ↔ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑦𝑧))
2423cbvrexvw 3383 . 2 (∃𝑤 ∈ ℝ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑤𝑧 ↔ ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑦𝑧)
2521, 24sylib 217 1 (𝜑 → ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑦𝑧)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  w3a 1086   = wceq 1539  wcel 2106  wral 3064  wrex 3065  Vcvv 3431   class class class wbr 5076  cmpt 5159  ran crn 5592  cr 10868  cle 11008
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5225  ax-nul 5232  ax-pr 5354
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3433  df-dif 3891  df-un 3893  df-in 3895  df-ss 3905  df-nul 4259  df-if 4462  df-sn 4564  df-pr 4566  df-op 4570  df-br 5077  df-opab 5139  df-mpt 5160  df-cnv 5599  df-dm 5601  df-rn 5602
This theorem is referenced by:  infnsuprnmpt  42766  infrpgernmpt  42975
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