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Theorem rnmptbd2lem 43723
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 2731 . . . . . . . 8 (𝑥𝐴𝐵) = (𝑥𝐴𝐵)
21elrnmpt 5947 . . . . . . 7 (𝑧 ∈ V → (𝑧 ∈ ran (𝑥𝐴𝐵) ↔ ∃𝑥𝐴 𝑧 = 𝐵))
32elv 3479 . . . . . 6 (𝑧 ∈ ran (𝑥𝐴𝐵) ↔ ∃𝑥𝐴 𝑧 = 𝐵)
4 nfra1 3280 . . . . . . . . 9 𝑥𝑥𝐴 𝑦𝐵
5 nfv 1917 . . . . . . . . 9 𝑥 𝑦𝑧
6 rspa 3244 . . . . . . . . . . 11 ((∀𝑥𝐴 𝑦𝐵𝑥𝐴) → 𝑦𝐵)
7 simpl 483 . . . . . . . . . . . . 13 ((𝑦𝐵𝑧 = 𝐵) → 𝑦𝐵)
8 id 22 . . . . . . . . . . . . . . 15 (𝑧 = 𝐵𝑧 = 𝐵)
98eqcomd 2737 . . . . . . . . . . . . . 14 (𝑧 = 𝐵𝐵 = 𝑧)
109adantl 482 . . . . . . . . . . . . 13 ((𝑦𝐵𝑧 = 𝐵) → 𝐵 = 𝑧)
117, 10breqtrd 5167 . . . . . . . . . . . 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 3262 . . . . . . . 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 3145 . . . 4 ((𝜑 ∧ ∀𝑥𝐴 𝑦𝐵) → ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑦𝑧)
2019ex 413 . . 3 (𝜑 → (∀𝑥𝐴 𝑦𝐵 → ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑦𝑧))
2120reximdv 3169 . 2 (𝜑 → (∃𝑦 ∈ ℝ ∀𝑥𝐴 𝑦𝐵 → ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑦𝑧))
22 rnmptbd2lem.x . . . . . 6 𝑥𝜑
23 nfmpt1 5249 . . . . . . . 8 𝑥(𝑥𝐴𝐵)
2423nfrn 5943 . . . . . . 7 𝑥ran (𝑥𝐴𝐵)
2524, 5nfralw 3307 . . . . . 6 𝑥𝑧 ∈ ran (𝑥𝐴𝐵)𝑦𝑧
2622, 25nfan 1902 . . . . 5 𝑥(𝜑 ∧ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑦𝑧)
27 breq2 5145 . . . . . 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 43702 . . . . . 6 (((𝜑 ∧ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑦𝑧) ∧ 𝑥𝐴) → 𝐵 ∈ ran (𝑥𝐴𝐵))
3327, 28, 32rspcdva 3610 . . . . 5 (((𝜑 ∧ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑦𝑧) ∧ 𝑥𝐴) → 𝑦𝐵)
3426, 33ralrimia 3254 . . . 4 ((𝜑 ∧ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑦𝑧) → ∀𝑥𝐴 𝑦𝐵)
3534ex 413 . . 3 (𝜑 → (∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑦𝑧 → ∀𝑥𝐴 𝑦𝐵))
3635reximdv 3169 . 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 3060  wrex 3069  Vcvv 3473   class class class wbr 5141  cmpt 5224  ran crn 5670  cr 11091  cle 11231
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 2702  ax-sep 5292  ax-nul 5299  ax-pr 5420
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 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4523  df-sn 4623  df-pr 4625  df-op 4629  df-br 5142  df-opab 5204  df-mpt 5225  df-cnv 5677  df-dm 5679  df-rn 5680
This theorem is referenced by:  rnmptbd2  43724
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