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Theorem rnmptbd 45100
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 5173 . . . . 5 (𝑦 = 𝑤 → (𝐵𝑦𝐵𝑤))
21ralbidv 3180 . . . 4 (𝑦 = 𝑤 → (∀𝑥𝐴 𝐵𝑦 ↔ ∀𝑥𝐴 𝐵𝑤))
32cbvrexvw 3239 . . 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 1913 . . 3 𝑤𝜑
7 rnmptbd.b . . 3 ((𝜑𝑥𝐴) → 𝐵𝑉)
85, 6, 7rnmptbdlem 45099 . 2 (𝜑 → (∃𝑤 ∈ ℝ ∀𝑥𝐴 𝐵𝑤 ↔ ∃𝑤 ∈ ℝ ∀𝑢 ∈ ran (𝑥𝐴𝐵)𝑢𝑤))
9 breq2 5173 . . . . . 6 (𝑤 = 𝑦 → (𝑢𝑤𝑢𝑦))
109ralbidv 3180 . . . . 5 (𝑤 = 𝑦 → (∀𝑢 ∈ ran (𝑥𝐴𝐵)𝑢𝑤 ↔ ∀𝑢 ∈ ran (𝑥𝐴𝐵)𝑢𝑦))
11 breq1 5172 . . . . . 6 (𝑢 = 𝑧 → (𝑢𝑦𝑧𝑦))
1211cbvralvw 3238 . . . . 5 (∀𝑢 ∈ ran (𝑥𝐴𝐵)𝑢𝑦 ↔ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝑦)
1310, 12bitrdi 287 . . . 4 (𝑤 = 𝑦 → (∀𝑢 ∈ ran (𝑥𝐴𝐵)𝑢𝑤 ↔ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝑦))
1413cbvrexvw 3239 . . 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 305 1 (𝜑 → (∃𝑦 ∈ ℝ ∀𝑥𝐴 𝐵𝑦 ↔ ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝑦))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wnf 1781  wcel 2103  wral 3063  wrex 3072   class class class wbr 5169  cmpt 5252  ran crn 5700  cr 11179  cle 11321
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2105  ax-9 2113  ax-10 2136  ax-11 2153  ax-12 2173  ax-ext 2705  ax-sep 5320  ax-nul 5327  ax-pr 5450
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2890  df-ral 3064  df-rex 3073  df-rab 3439  df-v 3484  df-sbc 3799  df-csb 3916  df-dif 3973  df-un 3975  df-ss 3987  df-nul 4348  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-br 5170  df-opab 5232  df-mpt 5253  df-cnv 5707  df-dm 5709  df-rn 5710
This theorem is referenced by:  supxrre3rnmpt  45279
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