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Theorem rnmptbdlem 43796
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 2903 . . . . . 6 𝑥
3 nfra1 3281 . . . . . 6 𝑥𝑥𝐴 𝐵𝑦
42, 3nfrexw 3310 . . . . 5 𝑥𝑦 ∈ ℝ ∀𝑥𝐴 𝐵𝑦
51, 4nfan 1902 . . . 4 𝑥(𝜑 ∧ ∃𝑦 ∈ ℝ ∀𝑥𝐴 𝐵𝑦)
6 simpr 485 . . . 4 ((𝜑 ∧ ∃𝑦 ∈ ℝ ∀𝑥𝐴 𝐵𝑦) → ∃𝑦 ∈ ℝ ∀𝑥𝐴 𝐵𝑦)
75, 6rnmptbdd 43785 . . 3 ((𝜑 ∧ ∃𝑦 ∈ ℝ ∀𝑥𝐴 𝐵𝑦) → ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝑦)
87ex 413 . 2 (𝜑 → (∃𝑦 ∈ ℝ ∀𝑥𝐴 𝐵𝑦 → ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝑦))
9 rnmptbdlem.y . . 3 𝑦𝜑
10 nfmpt1 5250 . . . . . . . . 9 𝑥(𝑥𝐴𝐵)
1110nfrn 5944 . . . . . . . 8 𝑥ran (𝑥𝐴𝐵)
12 nfv 1917 . . . . . . . 8 𝑥 𝑧𝑦
1311, 12nfralw 3308 . . . . . . 7 𝑥𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝑦
141, 13nfan 1902 . . . . . 6 𝑥(𝜑 ∧ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝑦)
15 breq1 5145 . . . . . . 7 (𝑧 = 𝐵 → (𝑧𝑦𝐵𝑦))
16 simplr 767 . . . . . . 7 (((𝜑 ∧ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝑦) ∧ 𝑥𝐴) → ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝑦)
17 eqid 2732 . . . . . . . 8 (𝑥𝐴𝐵) = (𝑥𝐴𝐵)
18 simpr 485 . . . . . . . 8 (((𝜑 ∧ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝑦) ∧ 𝑥𝐴) → 𝑥𝐴)
19 rnmptbdlem.b . . . . . . . . 9 ((𝜑𝑥𝐴) → 𝐵𝑉)
2019adantlr 713 . . . . . . . 8 (((𝜑 ∧ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝑦) ∧ 𝑥𝐴) → 𝐵𝑉)
2117, 18, 20elrnmpt1d 43768 . . . . . . 7 (((𝜑 ∧ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝑦) ∧ 𝑥𝐴) → 𝐵 ∈ ran (𝑥𝐴𝐵))
2215, 16, 21rspcdva 3611 . . . . . 6 (((𝜑 ∧ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝑦) ∧ 𝑥𝐴) → 𝐵𝑦)
2314, 22ralrimia 3255 . . . . 5 ((𝜑 ∧ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝑦) → ∀𝑥𝐴 𝐵𝑦)
2423ex 413 . . . 4 (𝜑 → (∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝑦 → ∀𝑥𝐴 𝐵𝑦))
2524a1d 25 . . 3 (𝜑 → (𝑦 ∈ ℝ → (∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝑦 → ∀𝑥𝐴 𝐵𝑦)))
269, 25reximdai 3258 . 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 396  wnf 1785  wcel 2106  wral 3061  wrex 3070   class class class wbr 5142  cmpt 5225  ran crn 5671  cr 11093  cle 11233
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5293  ax-nul 5300  ax-pr 5421
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-sbc 3775  df-csb 3891  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-nul 4320  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-br 5143  df-opab 5205  df-mpt 5226  df-cnv 5678  df-dm 5680  df-rn 5681
This theorem is referenced by:  rnmptbd  43797
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