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

Theorem rnmptbd2lem 43466
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 2736 . . . . . . . 8 (𝑥𝐴𝐵) = (𝑥𝐴𝐵)
21elrnmpt 5911 . . . . . . 7 (𝑧 ∈ V → (𝑧 ∈ ran (𝑥𝐴𝐵) ↔ ∃𝑥𝐴 𝑧 = 𝐵))
32elv 3451 . . . . . 6 (𝑧 ∈ ran (𝑥𝐴𝐵) ↔ ∃𝑥𝐴 𝑧 = 𝐵)
4 nfra1 3267 . . . . . . . . 9 𝑥𝑥𝐴 𝑦𝐵
5 nfv 1917 . . . . . . . . 9 𝑥 𝑦𝑧
6 rspa 3231 . . . . . . . . . . 11 ((∀𝑥𝐴 𝑦𝐵𝑥𝐴) → 𝑦𝐵)
7 simpl 483 . . . . . . . . . . . . 13 ((𝑦𝐵𝑧 = 𝐵) → 𝑦𝐵)
8 id 22 . . . . . . . . . . . . . . 15 (𝑧 = 𝐵𝑧 = 𝐵)
98eqcomd 2742 . . . . . . . . . . . . . 14 (𝑧 = 𝐵𝐵 = 𝑧)
109adantl 482 . . . . . . . . . . . . 13 ((𝑦𝐵𝑧 = 𝐵) → 𝐵 = 𝑧)
117, 10breqtrd 5131 . . . . . . . . . . . 12 ((𝑦𝐵𝑧 = 𝐵) → 𝑦𝑧)
1211ex 413 . . . . . . . . . . 11 (𝑦𝐵 → (𝑧 = 𝐵𝑦𝑧))
136, 12syl 17 . . . . . . . . . 10 ((∀𝑥𝐴 𝑦𝐵𝑥𝐴) → (𝑧 = 𝐵𝑦𝑧))
1413ex 413 . . . . . . . . 9 (∀𝑥𝐴 𝑦𝐵 → (𝑥𝐴 → (𝑧 = 𝐵𝑦𝑧)))
154, 5, 14rexlimd 3249 . . . . . . . 8 (∀𝑥𝐴 𝑦𝐵 → (∃𝑥𝐴 𝑧 = 𝐵𝑦𝑧))
1615imp 407 . . . . . . 7 ((∀𝑥𝐴 𝑦𝐵 ∧ ∃𝑥𝐴 𝑧 = 𝐵) → 𝑦𝑧)
1716adantll 712 . . . . . 6 (((𝜑 ∧ ∀𝑥𝐴 𝑦𝐵) ∧ ∃𝑥𝐴 𝑧 = 𝐵) → 𝑦𝑧)
183, 17sylan2b 594 . . . . 5 (((𝜑 ∧ ∀𝑥𝐴 𝑦𝐵) ∧ 𝑧 ∈ ran (𝑥𝐴𝐵)) → 𝑦𝑧)
1918ralrimiva 3143 . . . 4 ((𝜑 ∧ ∀𝑥𝐴 𝑦𝐵) → ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑦𝑧)
2019ex 413 . . 3 (𝜑 → (∀𝑥𝐴 𝑦𝐵 → ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑦𝑧))
2120reximdv 3167 . 2 (𝜑 → (∃𝑦 ∈ ℝ ∀𝑥𝐴 𝑦𝐵 → ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑦𝑧))
22 rnmptbd2lem.x . . . . . 6 𝑥𝜑
23 nfmpt1 5213 . . . . . . . 8 𝑥(𝑥𝐴𝐵)
2423nfrn 5907 . . . . . . 7 𝑥ran (𝑥𝐴𝐵)
2524, 5nfralw 3294 . . . . . 6 𝑥𝑧 ∈ ran (𝑥𝐴𝐵)𝑦𝑧
2622, 25nfan 1902 . . . . 5 𝑥(𝜑 ∧ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑦𝑧)
27 breq2 5109 . . . . . 6 (𝑧 = 𝐵 → (𝑦𝑧𝑦𝐵))
28 simplr 767 . . . . . 6 (((𝜑 ∧ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑦𝑧) ∧ 𝑥𝐴) → ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑦𝑧)
29 simpr 485 . . . . . . 7 (((𝜑 ∧ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑦𝑧) ∧ 𝑥𝐴) → 𝑥𝐴)
30 rnmptbd2lem.b . . . . . . . 8 ((𝜑𝑥𝐴) → 𝐵𝑉)
3130adantlr 713 . . . . . . 7 (((𝜑 ∧ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑦𝑧) ∧ 𝑥𝐴) → 𝐵𝑉)
321, 29, 31elrnmpt1d 43445 . . . . . 6 (((𝜑 ∧ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑦𝑧) ∧ 𝑥𝐴) → 𝐵 ∈ ran (𝑥𝐴𝐵))
3327, 28, 32rspcdva 3582 . . . . 5 (((𝜑 ∧ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑦𝑧) ∧ 𝑥𝐴) → 𝑦𝐵)
3426, 33ralrimia 3241 . . . 4 ((𝜑 ∧ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑦𝑧) → ∀𝑥𝐴 𝑦𝐵)
3534ex 413 . . 3 (𝜑 → (∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑦𝑧 → ∀𝑥𝐴 𝑦𝐵))
3635reximdv 3167 . 2 (𝜑 → (∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑦𝑧 → ∃𝑦 ∈ ℝ ∀𝑥𝐴 𝑦𝐵))
3721, 36impbid 211 1 (𝜑 → (∃𝑦 ∈ ℝ ∀𝑥𝐴 𝑦𝐵 ↔ ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑦𝑧))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1541  wnf 1785  wcel 2106  wral 3064  wrex 3073  Vcvv 3445   class class class wbr 5105  cmpt 5188  ran crn 5634  cr 11050  cle 11190
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 2707  ax-sep 5256  ax-nul 5263  ax-pr 5384
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 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ral 3065  df-rex 3074  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-sn 4587  df-pr 4589  df-op 4593  df-br 5106  df-opab 5168  df-mpt 5189  df-cnv 5641  df-dm 5643  df-rn 5644
This theorem is referenced by:  rnmptbd2  43467
  Copyright terms: Public domain W3C validator