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Theorem rnmptbdd 41807
 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 5056 . . . . . 6 (𝑦 = 𝑣 → (𝐵𝑦𝐵𝑣))
43ralbidv 3192 . . . . 5 (𝑦 = 𝑣 → (∀𝑥𝐴 𝐵𝑦 ↔ ∀𝑥𝐴 𝐵𝑣))
54cbvrexvw 3435 . . . 4 (∃𝑦 ∈ ℝ ∀𝑥𝐴 𝐵𝑦 ↔ ∃𝑣 ∈ ℝ ∀𝑥𝐴 𝐵𝑣)
62, 5sylib 221 . . 3 (𝜑 → ∃𝑣 ∈ ℝ ∀𝑥𝐴 𝐵𝑣)
71, 6rnmptbddlem 41806 . 2 (𝜑 → ∃𝑣 ∈ ℝ ∀𝑤 ∈ ran (𝑥𝐴𝐵)𝑤𝑣)
8 breq2 5056 . . . . 5 (𝑣 = 𝑦 → (𝑤𝑣𝑤𝑦))
98ralbidv 3192 . . . 4 (𝑣 = 𝑦 → (∀𝑤 ∈ ran (𝑥𝐴𝐵)𝑤𝑣 ↔ ∀𝑤 ∈ ran (𝑥𝐴𝐵)𝑤𝑦))
10 breq1 5055 . . . . 5 (𝑤 = 𝑧 → (𝑤𝑦𝑧𝑦))
1110cbvralvw 3434 . . . 4 (∀𝑤 ∈ ran (𝑥𝐴𝐵)𝑤𝑦 ↔ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝑦)
129, 11syl6bb 290 . . 3 (𝑣 = 𝑦 → (∀𝑤 ∈ ran (𝑥𝐴𝐵)𝑤𝑣 ↔ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝑦))
1312cbvrexvw 3435 . 2 (∃𝑣 ∈ ℝ ∀𝑤 ∈ ran (𝑥𝐴𝐵)𝑤𝑣 ↔ ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝑦)
147, 13sylib 221 1 (𝜑 → ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝑦)
 Colors of variables: wff setvar class Syntax hints:   → wi 4  Ⅎwnf 1785  ∀wral 3133  ∃wrex 3134   class class class wbr 5052   ↦ cmpt 5132  ran crn 5543  ℝcr 10534   ≤ cle 10674 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pr 5317 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3138  df-rex 3139  df-v 3482  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-sn 4551  df-pr 4553  df-op 4557  df-br 5053  df-opab 5115  df-mpt 5133  df-cnv 5550  df-dm 5552  df-rn 5553 This theorem is referenced by:  suprclrnmpt  41814  suprubrnmpt2  41815  suprubrnmpt  41816  rnmptbdlem  41818  supxrrernmpt  41984  suprleubrnmpt  41985
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