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Theorem rnmptbd 39983
Description: Boundness above of the range of a function in map-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 4788 . . . . 5 (𝑦 = 𝑤 → (𝐵𝑦𝐵𝑤))
21ralbidv 3134 . . . 4 (𝑦 = 𝑤 → (∀𝑥𝐴 𝐵𝑦 ↔ ∀𝑥𝐴 𝐵𝑤))
32cbvrexv 3320 . . 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 1994 . . 3 𝑤𝜑
7 rnmptbd.b . . 3 ((𝜑𝑥𝐴) → 𝐵𝑉)
85, 6, 7rnmptbdlem 39982 . 2 (𝜑 → (∃𝑤 ∈ ℝ ∀𝑥𝐴 𝐵𝑤 ↔ ∃𝑤 ∈ ℝ ∀𝑢 ∈ ran (𝑥𝐴𝐵)𝑢𝑤))
9 breq2 4788 . . . . . 6 (𝑤 = 𝑦 → (𝑢𝑤𝑢𝑦))
109ralbidv 3134 . . . . 5 (𝑤 = 𝑦 → (∀𝑢 ∈ ran (𝑥𝐴𝐵)𝑢𝑤 ↔ ∀𝑢 ∈ ran (𝑥𝐴𝐵)𝑢𝑦))
11 breq1 4787 . . . . . . 7 (𝑢 = 𝑧 → (𝑢𝑦𝑧𝑦))
1211cbvralv 3319 . . . . . 6 (∀𝑢 ∈ ran (𝑥𝐴𝐵)𝑢𝑦 ↔ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝑦)
1312a1i 11 . . . . 5 (𝑤 = 𝑦 → (∀𝑢 ∈ ran (𝑥𝐴𝐵)𝑢𝑦 ↔ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝑦))
1410, 13bitrd 268 . . . 4 (𝑤 = 𝑦 → (∀𝑢 ∈ ran (𝑥𝐴𝐵)𝑢𝑤 ↔ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝑦))
1514cbvrexv 3320 . . 3 (∃𝑤 ∈ ℝ ∀𝑢 ∈ ran (𝑥𝐴𝐵)𝑢𝑤 ↔ ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝑦)
1615a1i 11 . 2 (𝜑 → (∃𝑤 ∈ ℝ ∀𝑢 ∈ ran (𝑥𝐴𝐵)𝑢𝑤 ↔ ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝑦))
174, 8, 163bitrd 294 1 (𝜑 → (∃𝑦 ∈ ℝ ∀𝑥𝐴 𝐵𝑦 ↔ ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝑦))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382   = wceq 1630  wnf 1855  wcel 2144  wral 3060  wrex 3061   class class class wbr 4784  cmpt 4861  ran crn 5250  cr 10136  cle 10276
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-sep 4912  ax-nul 4920  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ral 3065  df-rex 3066  df-rab 3069  df-v 3351  df-sbc 3586  df-csb 3681  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-sn 4315  df-pr 4317  df-op 4321  df-br 4785  df-opab 4845  df-mpt 4862  df-cnv 5257  df-dm 5259  df-rn 5260
This theorem is referenced by:  supxrre3rnmpt  40166  supminfrnmpt  40182
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