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Theorem rnmptbdlem 43959
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 2904 . . . . . 6 𝑥
3 nfra1 3282 . . . . . 6 𝑥𝑥𝐴 𝐵𝑦
42, 3nfrexw 3311 . . . . 5 𝑥𝑦 ∈ ℝ ∀𝑥𝐴 𝐵𝑦
51, 4nfan 1903 . . . 4 𝑥(𝜑 ∧ ∃𝑦 ∈ ℝ ∀𝑥𝐴 𝐵𝑦)
6 simpr 486 . . . 4 ((𝜑 ∧ ∃𝑦 ∈ ℝ ∀𝑥𝐴 𝐵𝑦) → ∃𝑦 ∈ ℝ ∀𝑥𝐴 𝐵𝑦)
75, 6rnmptbdd 43949 . . 3 ((𝜑 ∧ ∃𝑦 ∈ ℝ ∀𝑥𝐴 𝐵𝑦) → ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝑦)
87ex 414 . 2 (𝜑 → (∃𝑦 ∈ ℝ ∀𝑥𝐴 𝐵𝑦 → ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝑦))
9 rnmptbdlem.y . . 3 𝑦𝜑
10 nfmpt1 5257 . . . . . . . . 9 𝑥(𝑥𝐴𝐵)
1110nfrn 5952 . . . . . . . 8 𝑥ran (𝑥𝐴𝐵)
12 nfv 1918 . . . . . . . 8 𝑥 𝑧𝑦
1311, 12nfralw 3309 . . . . . . 7 𝑥𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝑦
141, 13nfan 1903 . . . . . 6 𝑥(𝜑 ∧ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝑦)
15 breq1 5152 . . . . . . 7 (𝑧 = 𝐵 → (𝑧𝑦𝐵𝑦))
16 simplr 768 . . . . . . 7 (((𝜑 ∧ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝑦) ∧ 𝑥𝐴) → ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝑦)
17 eqid 2733 . . . . . . . 8 (𝑥𝐴𝐵) = (𝑥𝐴𝐵)
18 simpr 486 . . . . . . . 8 (((𝜑 ∧ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝑦) ∧ 𝑥𝐴) → 𝑥𝐴)
19 rnmptbdlem.b . . . . . . . . 9 ((𝜑𝑥𝐴) → 𝐵𝑉)
2019adantlr 714 . . . . . . . 8 (((𝜑 ∧ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝑦) ∧ 𝑥𝐴) → 𝐵𝑉)
2117, 18, 20elrnmpt1d 43932 . . . . . . 7 (((𝜑 ∧ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝑦) ∧ 𝑥𝐴) → 𝐵 ∈ ran (𝑥𝐴𝐵))
2215, 16, 21rspcdva 3614 . . . . . 6 (((𝜑 ∧ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝑦) ∧ 𝑥𝐴) → 𝐵𝑦)
2314, 22ralrimia 3256 . . . . 5 ((𝜑 ∧ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝑦) → ∀𝑥𝐴 𝐵𝑦)
2423ex 414 . . . 4 (𝜑 → (∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝑦 → ∀𝑥𝐴 𝐵𝑦))
2524a1d 25 . . 3 (𝜑 → (𝑦 ∈ ℝ → (∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝑦 → ∀𝑥𝐴 𝐵𝑦)))
269, 25reximdai 3259 . 2 (𝜑 → (∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝑦 → ∃𝑦 ∈ ℝ ∀𝑥𝐴 𝐵𝑦))
278, 26impbid 211 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 397  wnf 1786  wcel 2107  wral 3062  wrex 3071   class class class wbr 5149  cmpt 5232  ran crn 5678  cr 11109  cle 11249
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-br 5150  df-opab 5212  df-mpt 5233  df-cnv 5685  df-dm 5687  df-rn 5688
This theorem is referenced by:  rnmptbd  43960
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