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Theorem rnmptbdlem 41520
 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 2977 . . . . . 6 𝑥
3 nfra1 3219 . . . . . 6 𝑥𝑥𝐴 𝐵𝑦
42, 3nfrex 3309 . . . . 5 𝑥𝑦 ∈ ℝ ∀𝑥𝐴 𝐵𝑦
51, 4nfan 1896 . . . 4 𝑥(𝜑 ∧ ∃𝑦 ∈ ℝ ∀𝑥𝐴 𝐵𝑦)
6 simpr 487 . . . 4 ((𝜑 ∧ ∃𝑦 ∈ ℝ ∀𝑥𝐴 𝐵𝑦) → ∃𝑦 ∈ ℝ ∀𝑥𝐴 𝐵𝑦)
75, 6rnmptbdd 41509 . . 3 ((𝜑 ∧ ∃𝑦 ∈ ℝ ∀𝑥𝐴 𝐵𝑦) → ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝑦)
87ex 415 . 2 (𝜑 → (∃𝑦 ∈ ℝ ∀𝑥𝐴 𝐵𝑦 → ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝑦))
9 rnmptbdlem.y . . 3 𝑦𝜑
10 nfmpt1 5156 . . . . . . . . 9 𝑥(𝑥𝐴𝐵)
1110nfrn 5818 . . . . . . . 8 𝑥ran (𝑥𝐴𝐵)
12 nfv 1911 . . . . . . . 8 𝑥 𝑧𝑦
1311, 12nfralw 3225 . . . . . . 7 𝑥𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝑦
141, 13nfan 1896 . . . . . 6 𝑥(𝜑 ∧ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝑦)
15 breq1 5061 . . . . . . 7 (𝑧 = 𝐵 → (𝑧𝑦𝐵𝑦))
16 simplr 767 . . . . . . 7 (((𝜑 ∧ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝑦) ∧ 𝑥𝐴) → ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝑦)
17 eqid 2821 . . . . . . . 8 (𝑥𝐴𝐵) = (𝑥𝐴𝐵)
18 simpr 487 . . . . . . . 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 41493 . . . . . . 7 (((𝜑 ∧ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝑦) ∧ 𝑥𝐴) → 𝐵 ∈ ran (𝑥𝐴𝐵))
2215, 16, 21rspcdva 3624 . . . . . 6 (((𝜑 ∧ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝑦) ∧ 𝑥𝐴) → 𝐵𝑦)
2314, 22ralrimia 41391 . . . . 5 ((𝜑 ∧ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝑦) → ∀𝑥𝐴 𝐵𝑦)
2423ex 415 . . . 4 (𝜑 → (∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝑦 → ∀𝑥𝐴 𝐵𝑦))
2524a1d 25 . . 3 (𝜑 → (𝑦 ∈ ℝ → (∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝑦 → ∀𝑥𝐴 𝐵𝑦)))
269, 25reximdai 3311 . 2 (𝜑 → (∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝑦 → ∃𝑦 ∈ ℝ ∀𝑥𝐴 𝐵𝑦))
278, 26impbid 214 1 (𝜑 → (∃𝑦 ∈ ℝ ∀𝑥𝐴 𝐵𝑦 ↔ ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝑦))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398  Ⅎwnf 1780   ∈ wcel 2110  ∀wral 3138  ∃wrex 3139   class class class wbr 5058   ↦ cmpt 5138  ran crn 5550  ℝcr 10530   ≤ cle 10670 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5195  ax-nul 5202  ax-pr 5321 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-sn 4561  df-pr 4563  df-op 4567  df-br 5059  df-opab 5121  df-mpt 5139  df-cnv 5557  df-dm 5559  df-rn 5560 This theorem is referenced by:  rnmptbd  41521
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