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Theorem rnmptlb 41390
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 2818 . . . . . . 7 (𝑥𝐴𝐵) = (𝑥𝐴𝐵)
21elrnmpt 5821 . . . . . 6 (𝑧 ∈ V → (𝑧 ∈ ran (𝑥𝐴𝐵) ↔ ∃𝑥𝐴 𝑧 = 𝐵))
32elv 3497 . . . . 5 (𝑧 ∈ ran (𝑥𝐴𝐵) ↔ ∃𝑥𝐴 𝑧 = 𝐵)
4 nfra1 3216 . . . . . . . 8 𝑥𝑥𝐴 𝑤𝐵
5 nfv 1906 . . . . . . . 8 𝑥 𝑤𝑧
6 rspa 3203 . . . . . . . . . . 11 ((∀𝑥𝐴 𝑤𝐵𝑥𝐴) → 𝑤𝐵)
763adant3 1124 . . . . . . . . . 10 ((∀𝑥𝐴 𝑤𝐵𝑥𝐴𝑧 = 𝐵) → 𝑤𝐵)
8 simp3 1130 . . . . . . . . . 10 ((∀𝑥𝐴 𝑤𝐵𝑥𝐴𝑧 = 𝐵) → 𝑧 = 𝐵)
97, 8breqtrrd 5085 . . . . . . . . 9 ((∀𝑥𝐴 𝑤𝐵𝑥𝐴𝑧 = 𝐵) → 𝑤𝑧)
1093exp 1111 . . . . . . . 8 (∀𝑥𝐴 𝑤𝐵 → (𝑥𝐴 → (𝑧 = 𝐵𝑤𝑧)))
114, 5, 10rexlimd 3314 . . . . . . 7 (∀𝑥𝐴 𝑤𝐵 → (∃𝑥𝐴 𝑧 = 𝐵𝑤𝑧))
1211imp 407 . . . . . 6 ((∀𝑥𝐴 𝑤𝐵 ∧ ∃𝑥𝐴 𝑧 = 𝐵) → 𝑤𝑧)
1312adantll 710 . . . . 5 ((((𝜑𝑤 ∈ ℝ) ∧ ∀𝑥𝐴 𝑤𝐵) ∧ ∃𝑥𝐴 𝑧 = 𝐵) → 𝑤𝑧)
143, 13sylan2b 593 . . . 4 ((((𝜑𝑤 ∈ ℝ) ∧ ∀𝑥𝐴 𝑤𝐵) ∧ 𝑧 ∈ ran (𝑥𝐴𝐵)) → 𝑤𝑧)
1514ralrimiva 3179 . . 3 (((𝜑𝑤 ∈ ℝ) ∧ ∀𝑥𝐴 𝑤𝐵) → ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑤𝑧)
16 rnmptlb.1 . . . 4 (𝜑 → ∃𝑦 ∈ ℝ ∀𝑥𝐴 𝑦𝐵)
17 breq1 5060 . . . . . 6 (𝑦 = 𝑤 → (𝑦𝐵𝑤𝐵))
1817ralbidv 3194 . . . . 5 (𝑦 = 𝑤 → (∀𝑥𝐴 𝑦𝐵 ↔ ∀𝑥𝐴 𝑤𝐵))
1918cbvrexvw 3448 . . . 4 (∃𝑦 ∈ ℝ ∀𝑥𝐴 𝑦𝐵 ↔ ∃𝑤 ∈ ℝ ∀𝑥𝐴 𝑤𝐵)
2016, 19sylib 219 . . 3 (𝜑 → ∃𝑤 ∈ ℝ ∀𝑥𝐴 𝑤𝐵)
2115, 20reximddv3 41296 . 2 (𝜑 → ∃𝑤 ∈ ℝ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑤𝑧)
22 breq1 5060 . . . 4 (𝑤 = 𝑦 → (𝑤𝑧𝑦𝑧))
2322ralbidv 3194 . . 3 (𝑤 = 𝑦 → (∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑤𝑧 ↔ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑦𝑧))
2423cbvrexvw 3448 . 2 (∃𝑤 ∈ ℝ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑤𝑧 ↔ ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑦𝑧)
2521, 24sylib 219 1 (𝜑 → ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑦𝑧)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  w3a 1079   = wceq 1528  wcel 2105  wral 3135  wrex 3136  Vcvv 3492   class class class wbr 5057  cmpt 5137  ran crn 5549  cr 10524  cle 10664
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-br 5058  df-opab 5120  df-mpt 5138  df-cnv 5556  df-dm 5558  df-rn 5559
This theorem is referenced by:  infnsuprnmpt  41398  infrpgernmpt  41617
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