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Theorem rnmptbd 42784
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 5083 . . . . 5 (𝑦 = 𝑤 → (𝐵𝑦𝐵𝑤))
21ralbidv 3123 . . . 4 (𝑦 = 𝑤 → (∀𝑥𝐴 𝐵𝑦 ↔ ∀𝑥𝐴 𝐵𝑤))
32cbvrexvw 3382 . . 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 1921 . . 3 𝑤𝜑
7 rnmptbd.b . . 3 ((𝜑𝑥𝐴) → 𝐵𝑉)
85, 6, 7rnmptbdlem 42783 . 2 (𝜑 → (∃𝑤 ∈ ℝ ∀𝑥𝐴 𝐵𝑤 ↔ ∃𝑤 ∈ ℝ ∀𝑢 ∈ ran (𝑥𝐴𝐵)𝑢𝑤))
9 breq2 5083 . . . . . 6 (𝑤 = 𝑦 → (𝑢𝑤𝑢𝑦))
109ralbidv 3123 . . . . 5 (𝑤 = 𝑦 → (∀𝑢 ∈ ran (𝑥𝐴𝐵)𝑢𝑤 ↔ ∀𝑢 ∈ ran (𝑥𝐴𝐵)𝑢𝑦))
11 breq1 5082 . . . . . 6 (𝑢 = 𝑧 → (𝑢𝑦𝑧𝑦))
1211cbvralvw 3381 . . . . 5 (∀𝑢 ∈ ran (𝑥𝐴𝐵)𝑢𝑦 ↔ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝑦)
1310, 12bitrdi 287 . . . 4 (𝑤 = 𝑦 → (∀𝑢 ∈ ran (𝑥𝐴𝐵)𝑢𝑤 ↔ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝑦))
1413cbvrexvw 3382 . . 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 205  wa 396  wnf 1790  wcel 2110  wral 3066  wrex 3067   class class class wbr 5079  cmpt 5162  ran crn 5591  cr 10881  cle 11021
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2015  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2711  ax-sep 5227  ax-nul 5234  ax-pr 5356
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2072  df-mo 2542  df-eu 2571  df-clab 2718  df-cleq 2732  df-clel 2818  df-nfc 2891  df-ral 3071  df-rex 3072  df-rab 3075  df-v 3433  df-sbc 3721  df-csb 3838  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-nul 4263  df-if 4466  df-sn 4568  df-pr 4570  df-op 4574  df-br 5080  df-opab 5142  df-mpt 5163  df-cnv 5598  df-dm 5600  df-rn 5601
This theorem is referenced by:  supxrre3rnmpt  42951
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