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Theorem rnmptbd2lem 44037
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 2732 . . . . . . . 8 (𝑥𝐴𝐵) = (𝑥𝐴𝐵)
21elrnmpt 5955 . . . . . . 7 (𝑧 ∈ V → (𝑧 ∈ ran (𝑥𝐴𝐵) ↔ ∃𝑥𝐴 𝑧 = 𝐵))
32elv 3480 . . . . . 6 (𝑧 ∈ ran (𝑥𝐴𝐵) ↔ ∃𝑥𝐴 𝑧 = 𝐵)
4 nfra1 3281 . . . . . . . . 9 𝑥𝑥𝐴 𝑦𝐵
5 nfv 1917 . . . . . . . . 9 𝑥 𝑦𝑧
6 rspa 3245 . . . . . . . . . . 11 ((∀𝑥𝐴 𝑦𝐵𝑥𝐴) → 𝑦𝐵)
7 simpl 483 . . . . . . . . . . . . 13 ((𝑦𝐵𝑧 = 𝐵) → 𝑦𝐵)
8 id 22 . . . . . . . . . . . . . . 15 (𝑧 = 𝐵𝑧 = 𝐵)
98eqcomd 2738 . . . . . . . . . . . . . 14 (𝑧 = 𝐵𝐵 = 𝑧)
109adantl 482 . . . . . . . . . . . . 13 ((𝑦𝐵𝑧 = 𝐵) → 𝐵 = 𝑧)
117, 10breqtrd 5174 . . . . . . . . . . . 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 3263 . . . . . . . 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 3146 . . . 4 ((𝜑 ∧ ∀𝑥𝐴 𝑦𝐵) → ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑦𝑧)
2019ex 413 . . 3 (𝜑 → (∀𝑥𝐴 𝑦𝐵 → ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑦𝑧))
2120reximdv 3170 . 2 (𝜑 → (∃𝑦 ∈ ℝ ∀𝑥𝐴 𝑦𝐵 → ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑦𝑧))
22 rnmptbd2lem.x . . . . . 6 𝑥𝜑
23 nfmpt1 5256 . . . . . . . 8 𝑥(𝑥𝐴𝐵)
2423nfrn 5951 . . . . . . 7 𝑥ran (𝑥𝐴𝐵)
2524, 5nfralw 3308 . . . . . 6 𝑥𝑧 ∈ ran (𝑥𝐴𝐵)𝑦𝑧
2622, 25nfan 1902 . . . . 5 𝑥(𝜑 ∧ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑦𝑧)
27 breq2 5152 . . . . . 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 44017 . . . . . 6 (((𝜑 ∧ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑦𝑧) ∧ 𝑥𝐴) → 𝐵 ∈ ran (𝑥𝐴𝐵))
3327, 28, 32rspcdva 3613 . . . . 5 (((𝜑 ∧ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑦𝑧) ∧ 𝑥𝐴) → 𝑦𝐵)
3426, 33ralrimia 3255 . . . 4 ((𝜑 ∧ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑦𝑧) → ∀𝑥𝐴 𝑦𝐵)
3534ex 413 . . 3 (𝜑 → (∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑦𝑧 → ∀𝑥𝐴 𝑦𝐵))
3635reximdv 3170 . 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 3061  wrex 3070  Vcvv 3474   class class class wbr 5148  cmpt 5231  ran crn 5677  cr 11111  cle 11251
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 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427
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 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-br 5149  df-opab 5211  df-mpt 5232  df-cnv 5684  df-dm 5686  df-rn 5687
This theorem is referenced by:  rnmptbd2  44038
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