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Theorem rnmptbdlem 45262
Description: Boundness above of the range of a function in maps-to notation. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
Hypotheses
Ref Expression
rnmptbdlem.x 𝑥𝜑
rnmptbdlem.y 𝑦𝜑
rnmptbdlem.b ((𝜑𝑥𝐴) → 𝐵𝑉)
Assertion
Ref Expression
rnmptbdlem (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 rnmptbdlem
StepHypRef Expression
1 rnmptbdlem.x . . . . 5 𝑥𝜑
2 nfcv 2905 . . . . . 6 𝑥
3 nfra1 3284 . . . . . 6 𝑥𝑥𝐴 𝐵𝑦
42, 3nfrexw 3313 . . . . 5 𝑥𝑦 ∈ ℝ ∀𝑥𝐴 𝐵𝑦
51, 4nfan 1899 . . . 4 𝑥(𝜑 ∧ ∃𝑦 ∈ ℝ ∀𝑥𝐴 𝐵𝑦)
6 simpr 484 . . . 4 ((𝜑 ∧ ∃𝑦 ∈ ℝ ∀𝑥𝐴 𝐵𝑦) → ∃𝑦 ∈ ℝ ∀𝑥𝐴 𝐵𝑦)
75, 6rnmptbdd 45252 . . 3 ((𝜑 ∧ ∃𝑦 ∈ ℝ ∀𝑥𝐴 𝐵𝑦) → ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝑦)
87ex 412 . 2 (𝜑 → (∃𝑦 ∈ ℝ ∀𝑥𝐴 𝐵𝑦 → ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝑦))
9 rnmptbdlem.y . . 3 𝑦𝜑
10 nfmpt1 5250 . . . . . . . . 9 𝑥(𝑥𝐴𝐵)
1110nfrn 5963 . . . . . . . 8 𝑥ran (𝑥𝐴𝐵)
12 nfv 1914 . . . . . . . 8 𝑥 𝑧𝑦
1311, 12nfralw 3311 . . . . . . 7 𝑥𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝑦
141, 13nfan 1899 . . . . . 6 𝑥(𝜑 ∧ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝑦)
15 breq1 5146 . . . . . . 7 (𝑧 = 𝐵 → (𝑧𝑦𝐵𝑦))
16 simplr 769 . . . . . . 7 (((𝜑 ∧ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝑦) ∧ 𝑥𝐴) → ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝑦)
17 eqid 2737 . . . . . . . 8 (𝑥𝐴𝐵) = (𝑥𝐴𝐵)
18 simpr 484 . . . . . . . 8 (((𝜑 ∧ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝑦) ∧ 𝑥𝐴) → 𝑥𝐴)
19 rnmptbdlem.b . . . . . . . . 9 ((𝜑𝑥𝐴) → 𝐵𝑉)
2019adantlr 715 . . . . . . . 8 (((𝜑 ∧ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝑦) ∧ 𝑥𝐴) → 𝐵𝑉)
2117, 18, 20elrnmpt1d 5975 . . . . . . 7 (((𝜑 ∧ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝑦) ∧ 𝑥𝐴) → 𝐵 ∈ ran (𝑥𝐴𝐵))
2215, 16, 21rspcdva 3623 . . . . . 6 (((𝜑 ∧ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝑦) ∧ 𝑥𝐴) → 𝐵𝑦)
2314, 22ralrimia 3258 . . . . 5 ((𝜑 ∧ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝑦) → ∀𝑥𝐴 𝐵𝑦)
2423ex 412 . . . 4 (𝜑 → (∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝑦 → ∀𝑥𝐴 𝐵𝑦))
2524a1d 25 . . 3 (𝜑 → (𝑦 ∈ ℝ → (∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝑦 → ∀𝑥𝐴 𝐵𝑦)))
269, 25reximdai 3261 . 2 (𝜑 → (∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝑦 → ∃𝑦 ∈ ℝ ∀𝑥𝐴 𝐵𝑦))
278, 26impbid 212 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 1783  wcel 2108  wral 3061  wrex 3070   class class class wbr 5143  cmpt 5225  ran crn 5686  cr 11154  cle 11296
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-mpt 5226  df-cnv 5693  df-dm 5695  df-rn 5696
This theorem is referenced by:  rnmptbd  45263
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