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Theorem rnmptbd2 45160
Description: Boundness below of the range of a function in maps-to notation. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
Hypotheses
Ref Expression
rnmptbd2.x 𝑥𝜑
rnmptbd2.b ((𝜑𝑥𝐴) → 𝐵𝑉)
Assertion
Ref Expression
rnmptbd2 (𝜑 → (∃𝑦 ∈ ℝ ∀𝑥𝐴 𝑦𝐵 ↔ ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑦𝑧))
Distinct variable groups:   𝑦,𝐴,𝑧   𝑦,𝐵,𝑧   𝑥,𝑦,𝑧
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧)   𝐴(𝑥)   𝐵(𝑥)   𝑉(𝑥,𝑦,𝑧)

Proof of Theorem rnmptbd2
Dummy variables 𝑢 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 breq1 5169 . . . . 5 (𝑦 = 𝑤 → (𝑦𝐵𝑤𝐵))
21ralbidv 3184 . . . 4 (𝑦 = 𝑤 → (∀𝑥𝐴 𝑦𝐵 ↔ ∀𝑥𝐴 𝑤𝐵))
32cbvrexvw 3244 . . 3 (∃𝑦 ∈ ℝ ∀𝑥𝐴 𝑦𝐵 ↔ ∃𝑤 ∈ ℝ ∀𝑥𝐴 𝑤𝐵)
43a1i 11 . 2 (𝜑 → (∃𝑦 ∈ ℝ ∀𝑥𝐴 𝑦𝐵 ↔ ∃𝑤 ∈ ℝ ∀𝑥𝐴 𝑤𝐵))
5 rnmptbd2.x . . 3 𝑥𝜑
6 rnmptbd2.b . . 3 ((𝜑𝑥𝐴) → 𝐵𝑉)
75, 6rnmptbd2lem 45159 . 2 (𝜑 → (∃𝑤 ∈ ℝ ∀𝑥𝐴 𝑤𝐵 ↔ ∃𝑤 ∈ ℝ ∀𝑢 ∈ ran (𝑥𝐴𝐵)𝑤𝑢))
8 breq1 5169 . . . . . 6 (𝑤 = 𝑦 → (𝑤𝑢𝑦𝑢))
98ralbidv 3184 . . . . 5 (𝑤 = 𝑦 → (∀𝑢 ∈ ran (𝑥𝐴𝐵)𝑤𝑢 ↔ ∀𝑢 ∈ ran (𝑥𝐴𝐵)𝑦𝑢))
10 breq2 5170 . . . . . 6 (𝑢 = 𝑧 → (𝑦𝑢𝑦𝑧))
1110cbvralvw 3243 . . . . 5 (∀𝑢 ∈ ran (𝑥𝐴𝐵)𝑦𝑢 ↔ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑦𝑧)
129, 11bitrdi 287 . . . 4 (𝑤 = 𝑦 → (∀𝑢 ∈ ran (𝑥𝐴𝐵)𝑤𝑢 ↔ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑦𝑧))
1312cbvrexvw 3244 . . 3 (∃𝑤 ∈ ℝ ∀𝑢 ∈ ran (𝑥𝐴𝐵)𝑤𝑢 ↔ ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑦𝑧)
1413a1i 11 . 2 (𝜑 → (∃𝑤 ∈ ℝ ∀𝑢 ∈ ran (𝑥𝐴𝐵)𝑤𝑢 ↔ ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑦𝑧))
154, 7, 143bitrd 305 1 (𝜑 → (∃𝑦 ∈ ℝ ∀𝑥𝐴 𝑦𝐵 ↔ ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑦𝑧))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wnf 1781  wcel 2108  wral 3067  wrex 3076   class class class wbr 5166  cmpt 5249  ran crn 5701  cr 11185  cle 11327
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-br 5167  df-opab 5229  df-mpt 5250  df-cnv 5708  df-dm 5710  df-rn 5711
This theorem is referenced by:  limsupvaluz2  45661
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