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Theorem rnmptbdlem 39784
 Description: Boundness above of the range of a function in map-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 2793 . . . . . 6 𝑥
3 nfra1 2970 . . . . . 6 𝑥𝑥𝐴 𝐵𝑦
42, 3nfrex 3036 . . . . 5 𝑥𝑦 ∈ ℝ ∀𝑥𝐴 𝐵𝑦
51, 4nfan 1868 . . . 4 𝑥(𝜑 ∧ ∃𝑦 ∈ ℝ ∀𝑥𝐴 𝐵𝑦)
6 simpr 476 . . . 4 ((𝜑 ∧ ∃𝑦 ∈ ℝ ∀𝑥𝐴 𝐵𝑦) → ∃𝑦 ∈ ℝ ∀𝑥𝐴 𝐵𝑦)
75, 6rnmptbdd 39770 . . 3 ((𝜑 ∧ ∃𝑦 ∈ ℝ ∀𝑥𝐴 𝐵𝑦) → ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝑦)
87ex 449 . 2 (𝜑 → (∃𝑦 ∈ ℝ ∀𝑥𝐴 𝐵𝑦 → ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝑦))
9 rnmptbdlem.y . . 3 𝑦𝜑
10 nfmpt1 4780 . . . . . . . . 9 𝑥(𝑥𝐴𝐵)
1110nfrn 5400 . . . . . . . 8 𝑥ran (𝑥𝐴𝐵)
12 nfv 1883 . . . . . . . 8 𝑥 𝑧𝑦
1311, 12nfral 2974 . . . . . . 7 𝑥𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝑦
141, 13nfan 1868 . . . . . 6 𝑥(𝜑 ∧ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝑦)
15 simpr 476 . . . . . . . . 9 (((𝜑 ∧ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝑦) ∧ 𝑥𝐴) → 𝑥𝐴)
16 rnmptbdlem.b . . . . . . . . . 10 ((𝜑𝑥𝐴) → 𝐵𝑉)
1716adantlr 751 . . . . . . . . 9 (((𝜑 ∧ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝑦) ∧ 𝑥𝐴) → 𝐵𝑉)
18 eqid 2651 . . . . . . . . . 10 (𝑥𝐴𝐵) = (𝑥𝐴𝐵)
1918elrnmpt1 5406 . . . . . . . . 9 ((𝑥𝐴𝐵𝑉) → 𝐵 ∈ ran (𝑥𝐴𝐵))
2015, 17, 19syl2anc 694 . . . . . . . 8 (((𝜑 ∧ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝑦) ∧ 𝑥𝐴) → 𝐵 ∈ ran (𝑥𝐴𝐵))
21 simplr 807 . . . . . . . 8 (((𝜑 ∧ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝑦) ∧ 𝑥𝐴) → ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝑦)
22 breq1 4688 . . . . . . . . 9 (𝑧 = 𝐵 → (𝑧𝑦𝐵𝑦))
2322rspcva 3338 . . . . . . . 8 ((𝐵 ∈ ran (𝑥𝐴𝐵) ∧ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝑦) → 𝐵𝑦)
2420, 21, 23syl2anc 694 . . . . . . 7 (((𝜑 ∧ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝑦) ∧ 𝑥𝐴) → 𝐵𝑦)
2524ex 449 . . . . . 6 ((𝜑 ∧ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝑦) → (𝑥𝐴𝐵𝑦))
2614, 25ralrimi 2986 . . . . 5 ((𝜑 ∧ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝑦) → ∀𝑥𝐴 𝐵𝑦)
2726ex 449 . . . 4 (𝜑 → (∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝑦 → ∀𝑥𝐴 𝐵𝑦))
2827a1d 25 . . 3 (𝜑 → (𝑦 ∈ ℝ → (∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝑦 → ∀𝑥𝐴 𝐵𝑦)))
299, 28reximdai 3041 . 2 (𝜑 → (∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝑦 → ∃𝑦 ∈ ℝ ∀𝑥𝐴 𝐵𝑦))
308, 29impbid 202 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 383  Ⅎwnf 1748   ∈ wcel 2030  ∀wral 2941  ∃wrex 2942   class class class wbr 4685   ↦ cmpt 4762  ran crn 5144  ℝcr 9973   ≤ cle 10113 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-br 4686  df-opab 4746  df-mpt 4763  df-cnv 5151  df-dm 5153  df-rn 5154 This theorem is referenced by:  rnmptbd  39785
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