Users' Mathboxes Mathbox for Glauco Siliprandi < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  rnmptbd2 Structured version   Visualization version   GIF version

Theorem rnmptbd2 44252
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 5151 . . . . 5 (𝑦 = 𝑤 → (𝑦𝐵𝑤𝐵))
21ralbidv 3176 . . . 4 (𝑦 = 𝑤 → (∀𝑥𝐴 𝑦𝐵 ↔ ∀𝑥𝐴 𝑤𝐵))
32cbvrexvw 3234 . . 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 44251 . 2 (𝜑 → (∃𝑤 ∈ ℝ ∀𝑥𝐴 𝑤𝐵 ↔ ∃𝑤 ∈ ℝ ∀𝑢 ∈ ran (𝑥𝐴𝐵)𝑤𝑢))
8 breq1 5151 . . . . . 6 (𝑤 = 𝑦 → (𝑤𝑢𝑦𝑢))
98ralbidv 3176 . . . . 5 (𝑤 = 𝑦 → (∀𝑢 ∈ ran (𝑥𝐴𝐵)𝑤𝑢 ↔ ∀𝑢 ∈ ran (𝑥𝐴𝐵)𝑦𝑢))
10 breq2 5152 . . . . . 6 (𝑢 = 𝑧 → (𝑦𝑢𝑦𝑧))
1110cbvralvw 3233 . . . . 5 (∀𝑢 ∈ ran (𝑥𝐴𝐵)𝑦𝑢 ↔ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑦𝑧)
129, 11bitrdi 287 . . . 4 (𝑤 = 𝑦 → (∀𝑢 ∈ ran (𝑥𝐴𝐵)𝑤𝑢 ↔ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑦𝑧))
1312cbvrexvw 3234 . . 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 205  wa 395  wnf 1784  wcel 2105  wral 3060  wrex 3069   class class class wbr 5148  cmpt 5231  ran crn 5677  cr 11113  cle 11254
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-br 5149  df-opab 5211  df-mpt 5232  df-cnv 5684  df-dm 5686  df-rn 5687
This theorem is referenced by:  limsupvaluz2  44753
  Copyright terms: Public domain W3C validator