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Theorem rnmptbd2lem 44252
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 5956 . . . . . . 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 1916 . . . . . . . . 9 𝑥 𝑦𝑧
6 rspa 3244 . . . . . . . . . . 11 ((∀𝑥𝐴 𝑦𝐵𝑥𝐴) → 𝑦𝐵)
7 simpl 482 . . . . . . . . . . . . 13 ((𝑦𝐵𝑧 = 𝐵) → 𝑦𝐵)
8 id 22 . . . . . . . . . . . . . . 15 (𝑧 = 𝐵𝑧 = 𝐵)
98eqcomd 2737 . . . . . . . . . . . . . 14 (𝑧 = 𝐵𝐵 = 𝑧)
109adantl 481 . . . . . . . . . . . . 13 ((𝑦𝐵𝑧 = 𝐵) → 𝐵 = 𝑧)
117, 10breqtrd 5175 . . . . . . . . . . . 12 ((𝑦𝐵𝑧 = 𝐵) → 𝑦𝑧)
1211ex 412 . . . . . . . . . . 11 (𝑦𝐵 → (𝑧 = 𝐵𝑦𝑧))
136, 12syl 17 . . . . . . . . . 10 ((∀𝑥𝐴 𝑦𝐵𝑥𝐴) → (𝑧 = 𝐵𝑦𝑧))
1413ex 412 . . . . . . . . 9 (∀𝑥𝐴 𝑦𝐵 → (𝑥𝐴 → (𝑧 = 𝐵𝑦𝑧)))
154, 5, 14rexlimd 3262 . . . . . . . 8 (∀𝑥𝐴 𝑦𝐵 → (∃𝑥𝐴 𝑧 = 𝐵𝑦𝑧))
1615imp 406 . . . . . . 7 ((∀𝑥𝐴 𝑦𝐵 ∧ ∃𝑥𝐴 𝑧 = 𝐵) → 𝑦𝑧)
1716adantll 711 . . . . . 6 (((𝜑 ∧ ∀𝑥𝐴 𝑦𝐵) ∧ ∃𝑥𝐴 𝑧 = 𝐵) → 𝑦𝑧)
183, 17sylan2b 593 . . . . 5 (((𝜑 ∧ ∀𝑥𝐴 𝑦𝐵) ∧ 𝑧 ∈ ran (𝑥𝐴𝐵)) → 𝑦𝑧)
1918ralrimiva 3145 . . . 4 ((𝜑 ∧ ∀𝑥𝐴 𝑦𝐵) → ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑦𝑧)
2019ex 412 . . 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 5257 . . . . . . . 8 𝑥(𝑥𝐴𝐵)
2423nfrn 5952 . . . . . . 7 𝑥ran (𝑥𝐴𝐵)
2524, 5nfralw 3307 . . . . . 6 𝑥𝑧 ∈ ran (𝑥𝐴𝐵)𝑦𝑧
2622, 25nfan 1901 . . . . 5 𝑥(𝜑 ∧ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑦𝑧)
27 breq2 5153 . . . . . 6 (𝑧 = 𝐵 → (𝑦𝑧𝑦𝐵))
28 simplr 766 . . . . . 6 (((𝜑 ∧ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑦𝑧) ∧ 𝑥𝐴) → ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑦𝑧)
29 simpr 484 . . . . . . 7 (((𝜑 ∧ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑦𝑧) ∧ 𝑥𝐴) → 𝑥𝐴)
30 rnmptbd2lem.b . . . . . . . 8 ((𝜑𝑥𝐴) → 𝐵𝑉)
3130adantlr 712 . . . . . . 7 (((𝜑 ∧ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑦𝑧) ∧ 𝑥𝐴) → 𝐵𝑉)
321, 29, 31elrnmpt1d 44232 . . . . . 6 (((𝜑 ∧ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑦𝑧) ∧ 𝑥𝐴) → 𝐵 ∈ ran (𝑥𝐴𝐵))
3327, 28, 32rspcdva 3614 . . . . 5 (((𝜑 ∧ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑦𝑧) ∧ 𝑥𝐴) → 𝑦𝐵)
3426, 33ralrimia 3254 . . . 4 ((𝜑 ∧ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑦𝑧) → ∀𝑥𝐴 𝑦𝐵)
3534ex 412 . . 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 395   = wceq 1540  wnf 1784  wcel 2105  wral 3060  wrex 3069  Vcvv 3473   class class class wbr 5149  cmpt 5232  ran crn 5678  cr 11112  cle 11254
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5300  ax-nul 5307  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  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 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-br 5150  df-opab 5212  df-mpt 5233  df-cnv 5685  df-dm 5687  df-rn 5688
This theorem is referenced by:  rnmptbd2  44253
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