Mathbox for Glauco Siliprandi < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  rnmptbd2lem Structured version   Visualization version   GIF version

Theorem rnmptbd2lem 39777
 Description: Boundness below of the range of a function in map-to notation. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
Hypotheses
Ref Expression
rnmptbd2lem.x 𝑥𝜑
rnmptbd2lem.b ((𝜑𝑥𝐴) → 𝐵𝑉)
Assertion
Ref Expression
rnmptbd2lem (𝜑 → (∃𝑦 ∈ ℝ ∀𝑥𝐴 𝑦𝐵 ↔ ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑦𝑧))
Distinct variable groups:   𝑧,𝐴   𝑧,𝐵   𝜑,𝑦,𝑧   𝑥,𝑦,𝑧
Allowed substitution hints:   𝜑(𝑥)   𝐴(𝑥,𝑦)   𝐵(𝑥,𝑦)   𝑉(𝑥,𝑦,𝑧)

Proof of Theorem rnmptbd2lem
StepHypRef Expression
1 vex 3234 . . . . . . . . . . 11 𝑧 ∈ V
2 eqid 2651 . . . . . . . . . . . 12 (𝑥𝐴𝐵) = (𝑥𝐴𝐵)
32elrnmpt 5404 . . . . . . . . . . 11 (𝑧 ∈ V → (𝑧 ∈ ran (𝑥𝐴𝐵) ↔ ∃𝑥𝐴 𝑧 = 𝐵))
41, 3ax-mp 5 . . . . . . . . . 10 (𝑧 ∈ ran (𝑥𝐴𝐵) ↔ ∃𝑥𝐴 𝑧 = 𝐵)
54biimpi 206 . . . . . . . . 9 (𝑧 ∈ ran (𝑥𝐴𝐵) → ∃𝑥𝐴 𝑧 = 𝐵)
65adantl 481 . . . . . . . 8 (((𝜑 ∧ ∀𝑥𝐴 𝑦𝐵) ∧ 𝑧 ∈ ran (𝑥𝐴𝐵)) → ∃𝑥𝐴 𝑧 = 𝐵)
7 nfra1 2970 . . . . . . . . . . 11 𝑥𝑥𝐴 𝑦𝐵
8 nfv 1883 . . . . . . . . . . 11 𝑥 𝑦𝑧
9 rspa 2959 . . . . . . . . . . . . 13 ((∀𝑥𝐴 𝑦𝐵𝑥𝐴) → 𝑦𝐵)
10 simpl 472 . . . . . . . . . . . . . . 15 ((𝑦𝐵𝑧 = 𝐵) → 𝑦𝐵)
11 id 22 . . . . . . . . . . . . . . . . 17 (𝑧 = 𝐵𝑧 = 𝐵)
1211eqcomd 2657 . . . . . . . . . . . . . . . 16 (𝑧 = 𝐵𝐵 = 𝑧)
1312adantl 481 . . . . . . . . . . . . . . 15 ((𝑦𝐵𝑧 = 𝐵) → 𝐵 = 𝑧)
1410, 13breqtrd 4711 . . . . . . . . . . . . . 14 ((𝑦𝐵𝑧 = 𝐵) → 𝑦𝑧)
1514ex 449 . . . . . . . . . . . . 13 (𝑦𝐵 → (𝑧 = 𝐵𝑦𝑧))
169, 15syl 17 . . . . . . . . . . . 12 ((∀𝑥𝐴 𝑦𝐵𝑥𝐴) → (𝑧 = 𝐵𝑦𝑧))
1716ex 449 . . . . . . . . . . 11 (∀𝑥𝐴 𝑦𝐵 → (𝑥𝐴 → (𝑧 = 𝐵𝑦𝑧)))
187, 8, 17rexlimd 3055 . . . . . . . . . 10 (∀𝑥𝐴 𝑦𝐵 → (∃𝑥𝐴 𝑧 = 𝐵𝑦𝑧))
1918imp 444 . . . . . . . . 9 ((∀𝑥𝐴 𝑦𝐵 ∧ ∃𝑥𝐴 𝑧 = 𝐵) → 𝑦𝑧)
2019adantll 750 . . . . . . . 8 (((𝜑 ∧ ∀𝑥𝐴 𝑦𝐵) ∧ ∃𝑥𝐴 𝑧 = 𝐵) → 𝑦𝑧)
216, 20syldan 486 . . . . . . 7 (((𝜑 ∧ ∀𝑥𝐴 𝑦𝐵) ∧ 𝑧 ∈ ran (𝑥𝐴𝐵)) → 𝑦𝑧)
2221ralrimiva 2995 . . . . . 6 ((𝜑 ∧ ∀𝑥𝐴 𝑦𝐵) → ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑦𝑧)
2322ex 449 . . . . 5 (𝜑 → (∀𝑥𝐴 𝑦𝐵 → ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑦𝑧))
2423reximdv 3045 . . . 4 (𝜑 → (∃𝑦 ∈ ℝ ∀𝑥𝐴 𝑦𝐵 → ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑦𝑧))
2524imp 444 . . 3 ((𝜑 ∧ ∃𝑦 ∈ ℝ ∀𝑥𝐴 𝑦𝐵) → ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑦𝑧)
2625ex 449 . 2 (𝜑 → (∃𝑦 ∈ ℝ ∀𝑥𝐴 𝑦𝐵 → ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑦𝑧))
27 rnmptbd2lem.x . . . . . . . 8 𝑥𝜑
28 nfmpt1 4780 . . . . . . . . . 10 𝑥(𝑥𝐴𝐵)
2928nfrn 5400 . . . . . . . . 9 𝑥ran (𝑥𝐴𝐵)
3029, 8nfral 2974 . . . . . . . 8 𝑥𝑧 ∈ ran (𝑥𝐴𝐵)𝑦𝑧
3127, 30nfan 1868 . . . . . . 7 𝑥(𝜑 ∧ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑦𝑧)
32 simpr 476 . . . . . . . . . 10 (((𝜑 ∧ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑦𝑧) ∧ 𝑥𝐴) → 𝑥𝐴)
33 rnmptbd2lem.b . . . . . . . . . . 11 ((𝜑𝑥𝐴) → 𝐵𝑉)
3433adantlr 751 . . . . . . . . . 10 (((𝜑 ∧ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑦𝑧) ∧ 𝑥𝐴) → 𝐵𝑉)
352elrnmpt1 5406 . . . . . . . . . 10 ((𝑥𝐴𝐵𝑉) → 𝐵 ∈ ran (𝑥𝐴𝐵))
3632, 34, 35syl2anc 694 . . . . . . . . 9 (((𝜑 ∧ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑦𝑧) ∧ 𝑥𝐴) → 𝐵 ∈ ran (𝑥𝐴𝐵))
37 simplr 807 . . . . . . . . 9 (((𝜑 ∧ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑦𝑧) ∧ 𝑥𝐴) → ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑦𝑧)
38 breq2 4689 . . . . . . . . . 10 (𝑧 = 𝐵 → (𝑦𝑧𝑦𝐵))
3938rspcva 3338 . . . . . . . . 9 ((𝐵 ∈ ran (𝑥𝐴𝐵) ∧ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑦𝑧) → 𝑦𝐵)
4036, 37, 39syl2anc 694 . . . . . . . 8 (((𝜑 ∧ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑦𝑧) ∧ 𝑥𝐴) → 𝑦𝐵)
4140ex 449 . . . . . . 7 ((𝜑 ∧ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑦𝑧) → (𝑥𝐴𝑦𝐵))
4231, 41ralrimi 2986 . . . . . 6 ((𝜑 ∧ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑦𝑧) → ∀𝑥𝐴 𝑦𝐵)
4342ex 449 . . . . 5 (𝜑 → (∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑦𝑧 → ∀𝑥𝐴 𝑦𝐵))
4443a1d 25 . . . 4 (𝜑 → (𝑦 ∈ ℝ → (∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑦𝑧 → ∀𝑥𝐴 𝑦𝐵)))
4544imp 444 . . 3 ((𝜑𝑦 ∈ ℝ) → (∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑦𝑧 → ∀𝑥𝐴 𝑦𝐵))
4645reximdva 3046 . 2 (𝜑 → (∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑦𝑧 → ∃𝑦 ∈ ℝ ∀𝑥𝐴 𝑦𝐵))
4726, 46impbid 202 1 (𝜑 → (∃𝑦 ∈ ℝ ∀𝑥𝐴 𝑦𝐵 ↔ ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑦𝑧))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 383   = wceq 1523  Ⅎwnf 1748   ∈ wcel 2030  ∀wral 2941  ∃wrex 2942  Vcvv 3231   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:  rnmptbd2  39778
 Copyright terms: Public domain W3C validator