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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  Ⅎwnf 1785   ∈ wcel 2114  ∀wral 3130  ∃wrex 3131   class class class wbr 5042   ↦ cmpt 5122  ran crn 5533  ℝcr 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
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