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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  wnf 1748  wcel 2030  wral 2941  wrex 2942  Vcvv 3231   class class class wbr 4685  cmpt 4762  ran crn 5144  cr 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
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