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Theorem rnmptbdd 41392
Description: Boundness of the range of a function in maps-to notation. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
Hypotheses
Ref Expression
rnmptbdd.x 𝑥𝜑
rnmptbdd.b (𝜑 → ∃𝑦 ∈ ℝ ∀𝑥𝐴 𝐵𝑦)
Assertion
Ref Expression
rnmptbdd (𝜑 → ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝑦)
Distinct variable groups:   𝑦,𝐴,𝑧   𝑦,𝐵,𝑧   𝑥,𝑦,𝑧
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧)   𝐴(𝑥)   𝐵(𝑥)

Proof of Theorem rnmptbdd
Dummy variables 𝑣 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 rnmptbdd.x . . 3 𝑥𝜑
2 rnmptbdd.b . . . 4 (𝜑 → ∃𝑦 ∈ ℝ ∀𝑥𝐴 𝐵𝑦)
3 breq2 5061 . . . . . 6 (𝑦 = 𝑣 → (𝐵𝑦𝐵𝑣))
43ralbidv 3194 . . . . 5 (𝑦 = 𝑣 → (∀𝑥𝐴 𝐵𝑦 ↔ ∀𝑥𝐴 𝐵𝑣))
54cbvrexvw 3448 . . . 4 (∃𝑦 ∈ ℝ ∀𝑥𝐴 𝐵𝑦 ↔ ∃𝑣 ∈ ℝ ∀𝑥𝐴 𝐵𝑣)
62, 5sylib 219 . . 3 (𝜑 → ∃𝑣 ∈ ℝ ∀𝑥𝐴 𝐵𝑣)
71, 6rnmptbddlem 41391 . 2 (𝜑 → ∃𝑣 ∈ ℝ ∀𝑤 ∈ ran (𝑥𝐴𝐵)𝑤𝑣)
8 breq2 5061 . . . . 5 (𝑣 = 𝑦 → (𝑤𝑣𝑤𝑦))
98ralbidv 3194 . . . 4 (𝑣 = 𝑦 → (∀𝑤 ∈ ran (𝑥𝐴𝐵)𝑤𝑣 ↔ ∀𝑤 ∈ ran (𝑥𝐴𝐵)𝑤𝑦))
10 breq1 5060 . . . . 5 (𝑤 = 𝑧 → (𝑤𝑦𝑧𝑦))
1110cbvralvw 3447 . . . 4 (∀𝑤 ∈ ran (𝑥𝐴𝐵)𝑤𝑦 ↔ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝑦)
129, 11syl6bb 288 . . 3 (𝑣 = 𝑦 → (∀𝑤 ∈ ran (𝑥𝐴𝐵)𝑤𝑣 ↔ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝑦))
1312cbvrexvw 3448 . 2 (∃𝑣 ∈ ℝ ∀𝑤 ∈ ran (𝑥𝐴𝐵)𝑤𝑣 ↔ ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝑦)
147, 13sylib 219 1 (𝜑 → ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝑦)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wnf 1775  wral 3135  wrex 3136   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:  suprclrnmpt  41399  suprubrnmpt2  41400  suprubrnmpt  41401  rnmptbdlem  41403  supxrrernmpt  41571  suprleubrnmpt  41572
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