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Theorem rnmptlb 39767
Description: Boundness below of the range of a function in map-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 rnmptlb.1 . . . 4 (𝜑 → ∃𝑦 ∈ ℝ ∀𝑥𝐴 𝑦𝐵)
2 breq1 4688 . . . . . 6 (𝑦 = 𝑤 → (𝑦𝐵𝑤𝐵))
32ralbidv 3015 . . . . 5 (𝑦 = 𝑤 → (∀𝑥𝐴 𝑦𝐵 ↔ ∀𝑥𝐴 𝑤𝐵))
43cbvrexv 3202 . . . 4 (∃𝑦 ∈ ℝ ∀𝑥𝐴 𝑦𝐵 ↔ ∃𝑤 ∈ ℝ ∀𝑥𝐴 𝑤𝐵)
51, 4sylib 208 . . 3 (𝜑 → ∃𝑤 ∈ ℝ ∀𝑥𝐴 𝑤𝐵)
6 vex 3234 . . . . . . . . . . 11 𝑧 ∈ V
7 eqid 2651 . . . . . . . . . . . 12 (𝑥𝐴𝐵) = (𝑥𝐴𝐵)
87elrnmpt 5404 . . . . . . . . . . 11 (𝑧 ∈ V → (𝑧 ∈ ran (𝑥𝐴𝐵) ↔ ∃𝑥𝐴 𝑧 = 𝐵))
96, 8ax-mp 5 . . . . . . . . . 10 (𝑧 ∈ ran (𝑥𝐴𝐵) ↔ ∃𝑥𝐴 𝑧 = 𝐵)
109biimpi 206 . . . . . . . . 9 (𝑧 ∈ ran (𝑥𝐴𝐵) → ∃𝑥𝐴 𝑧 = 𝐵)
1110adantl 481 . . . . . . . 8 ((((𝜑𝑤 ∈ ℝ) ∧ ∀𝑥𝐴 𝑤𝐵) ∧ 𝑧 ∈ ran (𝑥𝐴𝐵)) → ∃𝑥𝐴 𝑧 = 𝐵)
12 nfra1 2970 . . . . . . . . . . 11 𝑥𝑥𝐴 𝑤𝐵
13 nfv 1883 . . . . . . . . . . 11 𝑥 𝑤𝑧
14 rspa 2959 . . . . . . . . . . . . . 14 ((∀𝑥𝐴 𝑤𝐵𝑥𝐴) → 𝑤𝐵)
15143adant3 1101 . . . . . . . . . . . . 13 ((∀𝑥𝐴 𝑤𝐵𝑥𝐴𝑧 = 𝐵) → 𝑤𝐵)
16 simp3 1083 . . . . . . . . . . . . 13 ((∀𝑥𝐴 𝑤𝐵𝑥𝐴𝑧 = 𝐵) → 𝑧 = 𝐵)
1715, 16breqtrrd 4713 . . . . . . . . . . . 12 ((∀𝑥𝐴 𝑤𝐵𝑥𝐴𝑧 = 𝐵) → 𝑤𝑧)
18173exp 1283 . . . . . . . . . . 11 (∀𝑥𝐴 𝑤𝐵 → (𝑥𝐴 → (𝑧 = 𝐵𝑤𝑧)))
1912, 13, 18rexlimd 3055 . . . . . . . . . 10 (∀𝑥𝐴 𝑤𝐵 → (∃𝑥𝐴 𝑧 = 𝐵𝑤𝑧))
2019imp 444 . . . . . . . . 9 ((∀𝑥𝐴 𝑤𝐵 ∧ ∃𝑥𝐴 𝑧 = 𝐵) → 𝑤𝑧)
2120adantll 750 . . . . . . . 8 ((((𝜑𝑤 ∈ ℝ) ∧ ∀𝑥𝐴 𝑤𝐵) ∧ ∃𝑥𝐴 𝑧 = 𝐵) → 𝑤𝑧)
2211, 21syldan 486 . . . . . . 7 ((((𝜑𝑤 ∈ ℝ) ∧ ∀𝑥𝐴 𝑤𝐵) ∧ 𝑧 ∈ ran (𝑥𝐴𝐵)) → 𝑤𝑧)
2322ralrimiva 2995 . . . . . 6 (((𝜑𝑤 ∈ ℝ) ∧ ∀𝑥𝐴 𝑤𝐵) → ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑤𝑧)
2423exp31 629 . . . . 5 (𝜑 → (𝑤 ∈ ℝ → (∀𝑥𝐴 𝑤𝐵 → ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑤𝑧)))
2524imp 444 . . . 4 ((𝜑𝑤 ∈ ℝ) → (∀𝑥𝐴 𝑤𝐵 → ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑤𝑧))
2625reximdva 3046 . . 3 (𝜑 → (∃𝑤 ∈ ℝ ∀𝑥𝐴 𝑤𝐵 → ∃𝑤 ∈ ℝ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑤𝑧))
275, 26mpd 15 . 2 (𝜑 → ∃𝑤 ∈ ℝ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑤𝑧)
28 breq1 4688 . . . 4 (𝑤 = 𝑦 → (𝑤𝑧𝑦𝑧))
2928ralbidv 3015 . . 3 (𝑤 = 𝑦 → (∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑤𝑧 ↔ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑦𝑧))
3029cbvrexv 3202 . 2 (∃𝑤 ∈ ℝ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑤𝑧 ↔ ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑦𝑧)
3127, 30sylib 208 1 (𝜑 → ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑦𝑧)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3a 1054   = wceq 1523  wcel 2030  wral 2941  wrex 2942  Vcvv 3231   class class class wbr 4685  cmpt 4762  ran crn 5144  cr 9973  cle 10113
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-br 4686  df-opab 4746  df-mpt 4763  df-cnv 5151  df-dm 5153  df-rn 5154
This theorem is referenced by:  infnsuprnmpt  39779  infrpgernmpt  40008
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