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Theorem rnmptbd2 39981
 Description: Boundness below of the range of a function in map-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 4807 . . . . 5 (𝑦 = 𝑤 → (𝑦𝐵𝑤𝐵))
21ralbidv 3124 . . . 4 (𝑦 = 𝑤 → (∀𝑥𝐴 𝑦𝐵 ↔ ∀𝑥𝐴 𝑤𝐵))
32cbvrexv 3311 . . 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 39980 . 2 (𝜑 → (∃𝑤 ∈ ℝ ∀𝑥𝐴 𝑤𝐵 ↔ ∃𝑤 ∈ ℝ ∀𝑢 ∈ ran (𝑥𝐴𝐵)𝑤𝑢))
8 breq1 4807 . . . . . 6 (𝑤 = 𝑦 → (𝑤𝑢𝑦𝑢))
98ralbidv 3124 . . . . 5 (𝑤 = 𝑦 → (∀𝑢 ∈ ran (𝑥𝐴𝐵)𝑤𝑢 ↔ ∀𝑢 ∈ ran (𝑥𝐴𝐵)𝑦𝑢))
10 breq2 4808 . . . . . . 7 (𝑢 = 𝑧 → (𝑦𝑢𝑦𝑧))
1110cbvralv 3310 . . . . . 6 (∀𝑢 ∈ ran (𝑥𝐴𝐵)𝑦𝑢 ↔ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑦𝑧)
1211a1i 11 . . . . 5 (𝑤 = 𝑦 → (∀𝑢 ∈ ran (𝑥𝐴𝐵)𝑦𝑢 ↔ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑦𝑧))
139, 12bitrd 268 . . . 4 (𝑤 = 𝑦 → (∀𝑢 ∈ ran (𝑥𝐴𝐵)𝑤𝑢 ↔ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑦𝑧))
1413cbvrexv 3311 . . 3 (∃𝑤 ∈ ℝ ∀𝑢 ∈ ran (𝑥𝐴𝐵)𝑤𝑢 ↔ ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑦𝑧)
1514a1i 11 . 2 (𝜑 → (∃𝑤 ∈ ℝ ∀𝑢 ∈ ran (𝑥𝐴𝐵)𝑤𝑢 ↔ ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑦𝑧))
164, 7, 153bitrd 294 1 (𝜑 → (∃𝑦 ∈ ℝ ∀𝑥𝐴 𝑦𝐵 ↔ ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑦𝑧))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 383   = wceq 1632  Ⅎwnf 1857   ∈ wcel 2139  ∀wral 3050  ∃wrex 3051   class class class wbr 4804   ↦ cmpt 4881  ran crn 5267  ℝcr 10147   ≤ cle 10287 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-br 4805  df-opab 4865  df-mpt 4882  df-cnv 5274  df-dm 5276  df-rn 5277 This theorem is referenced by:  limsupvaluz2  40491
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