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

Theorem rnmptbd2lem 42239
 Description: Boundness below of the range of a function in maps-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 eqid 2759 . . . . . . . 8 (𝑥𝐴𝐵) = (𝑥𝐴𝐵)
21elrnmpt 5790 . . . . . . 7 (𝑧 ∈ V → (𝑧 ∈ ran (𝑥𝐴𝐵) ↔ ∃𝑥𝐴 𝑧 = 𝐵))
32elv 3413 . . . . . 6 (𝑧 ∈ ran (𝑥𝐴𝐵) ↔ ∃𝑥𝐴 𝑧 = 𝐵)
4 nfra1 3145 . . . . . . . . 9 𝑥𝑥𝐴 𝑦𝐵
5 nfv 1916 . . . . . . . . 9 𝑥 𝑦𝑧
6 rspa 3133 . . . . . . . . . . 11 ((∀𝑥𝐴 𝑦𝐵𝑥𝐴) → 𝑦𝐵)
7 simpl 487 . . . . . . . . . . . . 13 ((𝑦𝐵𝑧 = 𝐵) → 𝑦𝐵)
8 id 22 . . . . . . . . . . . . . . 15 (𝑧 = 𝐵𝑧 = 𝐵)
98eqcomd 2765 . . . . . . . . . . . . . 14 (𝑧 = 𝐵𝐵 = 𝑧)
109adantl 486 . . . . . . . . . . . . 13 ((𝑦𝐵𝑧 = 𝐵) → 𝐵 = 𝑧)
117, 10breqtrd 5051 . . . . . . . . . . . 12 ((𝑦𝐵𝑧 = 𝐵) → 𝑦𝑧)
1211ex 417 . . . . . . . . . . 11 (𝑦𝐵 → (𝑧 = 𝐵𝑦𝑧))
136, 12syl 17 . . . . . . . . . 10 ((∀𝑥𝐴 𝑦𝐵𝑥𝐴) → (𝑧 = 𝐵𝑦𝑧))
1413ex 417 . . . . . . . . 9 (∀𝑥𝐴 𝑦𝐵 → (𝑥𝐴 → (𝑧 = 𝐵𝑦𝑧)))
154, 5, 14rexlimd 3239 . . . . . . . 8 (∀𝑥𝐴 𝑦𝐵 → (∃𝑥𝐴 𝑧 = 𝐵𝑦𝑧))
1615imp 411 . . . . . . 7 ((∀𝑥𝐴 𝑦𝐵 ∧ ∃𝑥𝐴 𝑧 = 𝐵) → 𝑦𝑧)
1716adantll 714 . . . . . 6 (((𝜑 ∧ ∀𝑥𝐴 𝑦𝐵) ∧ ∃𝑥𝐴 𝑧 = 𝐵) → 𝑦𝑧)
183, 17sylan2b 597 . . . . 5 (((𝜑 ∧ ∀𝑥𝐴 𝑦𝐵) ∧ 𝑧 ∈ ran (𝑥𝐴𝐵)) → 𝑦𝑧)
1918ralrimiva 3111 . . . 4 ((𝜑 ∧ ∀𝑥𝐴 𝑦𝐵) → ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑦𝑧)
2019ex 417 . . 3 (𝜑 → (∀𝑥𝐴 𝑦𝐵 → ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑦𝑧))
2120reximdv 3195 . 2 (𝜑 → (∃𝑦 ∈ ℝ ∀𝑥𝐴 𝑦𝐵 → ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑦𝑧))
22 rnmptbd2lem.x . . . . . 6 𝑥𝜑
23 nfmpt1 5123 . . . . . . . 8 𝑥(𝑥𝐴𝐵)
2423nfrn 5786 . . . . . . 7 𝑥ran (𝑥𝐴𝐵)
2524, 5nfralw 3151 . . . . . 6 𝑥𝑧 ∈ ran (𝑥𝐴𝐵)𝑦𝑧
2622, 25nfan 1901 . . . . 5 𝑥(𝜑 ∧ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑦𝑧)
27 breq2 5029 . . . . . 6 (𝑧 = 𝐵 → (𝑦𝑧𝑦𝐵))
28 simplr 769 . . . . . 6 (((𝜑 ∧ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑦𝑧) ∧ 𝑥𝐴) → ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑦𝑧)
29 simpr 489 . . . . . . 7 (((𝜑 ∧ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑦𝑧) ∧ 𝑥𝐴) → 𝑥𝐴)
30 rnmptbd2lem.b . . . . . . . 8 ((𝜑𝑥𝐴) → 𝐵𝑉)
3130adantlr 715 . . . . . . 7 (((𝜑 ∧ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑦𝑧) ∧ 𝑥𝐴) → 𝐵𝑉)
321, 29, 31elrnmpt1d 42219 . . . . . 6 (((𝜑 ∧ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑦𝑧) ∧ 𝑥𝐴) → 𝐵 ∈ ran (𝑥𝐴𝐵))
3327, 28, 32rspcdva 3541 . . . . 5 (((𝜑 ∧ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑦𝑧) ∧ 𝑥𝐴) → 𝑦𝐵)
3426, 33ralrimia 3405 . . . 4 ((𝜑 ∧ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑦𝑧) → ∀𝑥𝐴 𝑦𝐵)
3534ex 417 . . 3 (𝜑 → (∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑦𝑧 → ∀𝑥𝐴 𝑦𝐵))
3635reximdv 3195 . 2 (𝜑 → (∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑦𝑧 → ∃𝑦 ∈ ℝ ∀𝑥𝐴 𝑦𝐵))
3721, 36impbid 215 1 (𝜑 → (∃𝑦 ∈ ℝ ∀𝑥𝐴 𝑦𝐵 ↔ ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑦𝑧))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1539  Ⅎwnf 1786   ∈ wcel 2112  ∀wral 3068  ∃wrex 3069  Vcvv 3407   class class class wbr 5025   ↦ cmpt 5105  ran crn 5518  ℝcr 10559   ≤ cle 10699 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 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2730  ax-sep 5162  ax-nul 5169  ax-pr 5291 This theorem depends on definitions:  df-bi 210  df-an 401  df-or 846  df-3an 1087  df-tru 1542  df-ex 1783  df-nf 1787  df-sb 2071  df-mo 2558  df-eu 2589  df-clab 2737  df-cleq 2751  df-clel 2831  df-nfc 2899  df-ral 3073  df-rex 3074  df-v 3409  df-sbc 3694  df-csb 3802  df-dif 3857  df-un 3859  df-in 3861  df-ss 3871  df-nul 4222  df-if 4414  df-sn 4516  df-pr 4518  df-op 4522  df-br 5026  df-opab 5088  df-mpt 5106  df-cnv 5525  df-dm 5527  df-rn 5528 This theorem is referenced by:  rnmptbd2  42240
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