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

Theorem rnmptbd 41832
Description: Boundness above of the range of a function in maps-to notation. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
Hypotheses
Ref Expression
rnmptbd.x 𝑥𝜑
rnmptbd.b ((𝜑𝑥𝐴) → 𝐵𝑉)
Assertion
Ref Expression
rnmptbd (𝜑 → (∃𝑦 ∈ ℝ ∀𝑥𝐴 𝐵𝑦 ↔ ∃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)   𝑉(𝑥,𝑦,𝑧)

Proof of Theorem rnmptbd
Dummy variables 𝑢 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 breq2 5046 . . . . 5 (𝑦 = 𝑤 → (𝐵𝑦𝐵𝑤))
21ralbidv 3187 . . . 4 (𝑦 = 𝑤 → (∀𝑥𝐴 𝐵𝑦 ↔ ∀𝑥𝐴 𝐵𝑤))
32cbvrexvw 3425 . . 3 (∃𝑦 ∈ ℝ ∀𝑥𝐴 𝐵𝑦 ↔ ∃𝑤 ∈ ℝ ∀𝑥𝐴 𝐵𝑤)
43a1i 11 . 2 (𝜑 → (∃𝑦 ∈ ℝ ∀𝑥𝐴 𝐵𝑦 ↔ ∃𝑤 ∈ ℝ ∀𝑥𝐴 𝐵𝑤))
5 rnmptbd.x . . 3 𝑥𝜑
6 nfv 1915 . . 3 𝑤𝜑
7 rnmptbd.b . . 3 ((𝜑𝑥𝐴) → 𝐵𝑉)
85, 6, 7rnmptbdlem 41831 . 2 (𝜑 → (∃𝑤 ∈ ℝ ∀𝑥𝐴 𝐵𝑤 ↔ ∃𝑤 ∈ ℝ ∀𝑢 ∈ ran (𝑥𝐴𝐵)𝑢𝑤))
9 breq2 5046 . . . . . 6 (𝑤 = 𝑦 → (𝑢𝑤𝑢𝑦))
109ralbidv 3187 . . . . 5 (𝑤 = 𝑦 → (∀𝑢 ∈ ran (𝑥𝐴𝐵)𝑢𝑤 ↔ ∀𝑢 ∈ ran (𝑥𝐴𝐵)𝑢𝑦))
11 breq1 5045 . . . . . 6 (𝑢 = 𝑧 → (𝑢𝑦𝑧𝑦))
1211cbvralvw 3424 . . . . 5 (∀𝑢 ∈ ran (𝑥𝐴𝐵)𝑢𝑦 ↔ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝑦)
1310, 12syl6bb 290 . . . 4 (𝑤 = 𝑦 → (∀𝑢 ∈ ran (𝑥𝐴𝐵)𝑢𝑤 ↔ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝑦))
1413cbvrexvw 3425 . . 3 (∃𝑤 ∈ ℝ ∀𝑢 ∈ ran (𝑥𝐴𝐵)𝑢𝑤 ↔ ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝑦)
1514a1i 11 . 2 (𝜑 → (∃𝑤 ∈ ℝ ∀𝑢 ∈ ran (𝑥𝐴𝐵)𝑢𝑤 ↔ ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝑦))
164, 8, 153bitrd 308 1 (𝜑 → (∃𝑦 ∈ ℝ ∀𝑥𝐴 𝐵𝑦 ↔ ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝑦))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  wnf 1785  wcel 2114  wral 3130  wrex 3131   class class class wbr 5042  cmpt 5122  ran crn 5533  cr 10525  cle 10665
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 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-sep 5179  ax-nul 5186  ax-pr 5307
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 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ral 3135  df-rex 3136  df-v 3471  df-sbc 3748  df-csb 3856  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-sn 4540  df-pr 4542  df-op 4546  df-br 5043  df-opab 5105  df-mpt 5123  df-cnv 5540  df-dm 5542  df-rn 5543
This theorem is referenced by:  supxrre3rnmpt  42005  supminfrnmpt  42021
  Copyright terms: Public domain W3C validator