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Theorem rnmptbdd 40258
 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 4878 . . . . . 6 (𝑦 = 𝑣 → (𝐵𝑦𝐵𝑣))
43ralbidv 3196 . . . . 5 (𝑦 = 𝑣 → (∀𝑥𝐴 𝐵𝑦 ↔ ∀𝑥𝐴 𝐵𝑣))
54cbvrexv 3385 . . . 4 (∃𝑦 ∈ ℝ ∀𝑥𝐴 𝐵𝑦 ↔ ∃𝑣 ∈ ℝ ∀𝑥𝐴 𝐵𝑣)
62, 5sylib 210 . . 3 (𝜑 → ∃𝑣 ∈ ℝ ∀𝑥𝐴 𝐵𝑣)
71, 6rnmptbddlem 40257 . 2 (𝜑 → ∃𝑣 ∈ ℝ ∀𝑤 ∈ ran (𝑥𝐴𝐵)𝑤𝑣)
8 breq2 4878 . . . . 5 (𝑣 = 𝑦 → (𝑤𝑣𝑤𝑦))
98ralbidv 3196 . . . 4 (𝑣 = 𝑦 → (∀𝑤 ∈ ran (𝑥𝐴𝐵)𝑤𝑣 ↔ ∀𝑤 ∈ ran (𝑥𝐴𝐵)𝑤𝑦))
10 breq1 4877 . . . . . 6 (𝑤 = 𝑧 → (𝑤𝑦𝑧𝑦))
1110cbvralv 3384 . . . . 5 (∀𝑤 ∈ ran (𝑥𝐴𝐵)𝑤𝑦 ↔ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝑦)
1211a1i 11 . . . 4 (𝑣 = 𝑦 → (∀𝑤 ∈ ran (𝑥𝐴𝐵)𝑤𝑦 ↔ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝑦))
139, 12bitrd 271 . . 3 (𝑣 = 𝑦 → (∀𝑤 ∈ ran (𝑥𝐴𝐵)𝑤𝑣 ↔ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝑦))
1413cbvrexv 3385 . 2 (∃𝑣 ∈ ℝ ∀𝑤 ∈ ran (𝑥𝐴𝐵)𝑤𝑣 ↔ ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝑦)
157, 14sylib 210 1 (𝜑 → ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝑦)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 198   = wceq 1658  Ⅎwnf 1884  ∀wral 3118  ∃wrex 3119   class class class wbr 4874   ↦ cmpt 4953  ran crn 5344  ℝcr 10252   ≤ cle 10393 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2804  ax-sep 5006  ax-nul 5014  ax-pr 5128 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2606  df-eu 2641  df-clab 2813  df-cleq 2819  df-clel 2822  df-nfc 2959  df-ral 3123  df-rex 3124  df-rab 3127  df-v 3417  df-dif 3802  df-un 3804  df-in 3806  df-ss 3813  df-nul 4146  df-if 4308  df-sn 4399  df-pr 4401  df-op 4405  df-br 4875  df-opab 4937  df-mpt 4954  df-cnv 5351  df-dm 5353  df-rn 5354 This theorem is referenced by:  suprclrnmpt  40267  suprubrnmpt2  40268  suprubrnmpt  40269  rnmptbdlem  40271  supxrrernmpt  40444  suprleubrnmpt  40445
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