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Theorem infxrunb3rnmpt 41856
Description: The infimum of an unbounded-below set of extended reals is minus infinity. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
Hypotheses
Ref Expression
infxrunb3rnmpt.1 𝑥𝜑
infxrunb3rnmpt.2 𝑦𝜑
infxrunb3rnmpt.3 ((𝜑𝑥𝐴) → 𝐵 ∈ ℝ*)
Assertion
Ref Expression
infxrunb3rnmpt (𝜑 → (∀𝑦 ∈ ℝ ∃𝑥𝐴 𝐵𝑦 ↔ inf(ran (𝑥𝐴𝐵), ℝ*, < ) = -∞))
Distinct variable groups:   𝑥,𝐴,𝑦   𝑦,𝐵
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐵(𝑥)

Proof of Theorem infxrunb3rnmpt
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 infxrunb3rnmpt.2 . . 3 𝑦𝜑
2 infxrunb3rnmpt.1 . . . . 5 𝑥𝜑
3 nfmpt1 5137 . . . . . . 7 𝑥(𝑥𝐴𝐵)
43nfrn 5797 . . . . . 6 𝑥ran (𝑥𝐴𝐵)
5 nfv 1916 . . . . . 6 𝑥 𝑧𝑦
64, 5nfrex 3295 . . . . 5 𝑥𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝑦
7 simpr 488 . . . . . . . . 9 ((𝜑𝑥𝐴) → 𝑥𝐴)
8 infxrunb3rnmpt.3 . . . . . . . . 9 ((𝜑𝑥𝐴) → 𝐵 ∈ ℝ*)
9 eqid 2821 . . . . . . . . . 10 (𝑥𝐴𝐵) = (𝑥𝐴𝐵)
109elrnmpt1 5803 . . . . . . . . 9 ((𝑥𝐴𝐵 ∈ ℝ*) → 𝐵 ∈ ran (𝑥𝐴𝐵))
117, 8, 10syl2anc 587 . . . . . . . 8 ((𝜑𝑥𝐴) → 𝐵 ∈ ran (𝑥𝐴𝐵))
12113adant3 1129 . . . . . . 7 ((𝜑𝑥𝐴𝐵𝑦) → 𝐵 ∈ ran (𝑥𝐴𝐵))
13 simp3 1135 . . . . . . 7 ((𝜑𝑥𝐴𝐵𝑦) → 𝐵𝑦)
14 breq1 5042 . . . . . . . 8 (𝑧 = 𝐵 → (𝑧𝑦𝐵𝑦))
1514rspcev 3600 . . . . . . 7 ((𝐵 ∈ ran (𝑥𝐴𝐵) ∧ 𝐵𝑦) → ∃𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝑦)
1612, 13, 15syl2anc 587 . . . . . 6 ((𝜑𝑥𝐴𝐵𝑦) → ∃𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝑦)
17163exp 1116 . . . . 5 (𝜑 → (𝑥𝐴 → (𝐵𝑦 → ∃𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝑦)))
182, 6, 17rexlimd 3303 . . . 4 (𝜑 → (∃𝑥𝐴 𝐵𝑦 → ∃𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝑦))
19 nfv 1916 . . . . . 6 𝑧𝑥𝐴 𝐵𝑦
20 vex 3474 . . . . . . . . 9 𝑧 ∈ V
219elrnmpt 5801 . . . . . . . . 9 (𝑧 ∈ V → (𝑧 ∈ ran (𝑥𝐴𝐵) ↔ ∃𝑥𝐴 𝑧 = 𝐵))
2220, 21ax-mp 5 . . . . . . . 8 (𝑧 ∈ ran (𝑥𝐴𝐵) ↔ ∃𝑥𝐴 𝑧 = 𝐵)
2322biimpi 219 . . . . . . 7 (𝑧 ∈ ran (𝑥𝐴𝐵) → ∃𝑥𝐴 𝑧 = 𝐵)
2414biimpcd 252 . . . . . . . . . 10 (𝑧𝑦 → (𝑧 = 𝐵𝐵𝑦))
2524a1d 25 . . . . . . . . 9 (𝑧𝑦 → (𝑥𝐴 → (𝑧 = 𝐵𝐵𝑦)))
265, 25reximdai 3297 . . . . . . . 8 (𝑧𝑦 → (∃𝑥𝐴 𝑧 = 𝐵 → ∃𝑥𝐴 𝐵𝑦))
2726com12 32 . . . . . . 7 (∃𝑥𝐴 𝑧 = 𝐵 → (𝑧𝑦 → ∃𝑥𝐴 𝐵𝑦))
2823, 27syl 17 . . . . . 6 (𝑧 ∈ ran (𝑥𝐴𝐵) → (𝑧𝑦 → ∃𝑥𝐴 𝐵𝑦))
2919, 28rexlimi 3301 . . . . 5 (∃𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝑦 → ∃𝑥𝐴 𝐵𝑦)
3029a1i 11 . . . 4 (𝜑 → (∃𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝑦 → ∃𝑥𝐴 𝐵𝑦))
3118, 30impbid 215 . . 3 (𝜑 → (∃𝑥𝐴 𝐵𝑦 ↔ ∃𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝑦))
321, 31ralbid 3219 . 2 (𝜑 → (∀𝑦 ∈ ℝ ∃𝑥𝐴 𝐵𝑦 ↔ ∀𝑦 ∈ ℝ ∃𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝑦))
332, 9, 8rnmptssd 41612 . . 3 (𝜑 → ran (𝑥𝐴𝐵) ⊆ ℝ*)
34 infxrunb3 41852 . . 3 (ran (𝑥𝐴𝐵) ⊆ ℝ* → (∀𝑦 ∈ ℝ ∃𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝑦 ↔ inf(ran (𝑥𝐴𝐵), ℝ*, < ) = -∞))
3533, 34syl 17 . 2 (𝜑 → (∀𝑦 ∈ ℝ ∃𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝑦 ↔ inf(ran (𝑥𝐴𝐵), ℝ*, < ) = -∞))
3632, 35bitrd 282 1 (𝜑 → (∀𝑦 ∈ ℝ ∃𝑥𝐴 𝐵𝑦 ↔ inf(ran (𝑥𝐴𝐵), ℝ*, < ) = -∞))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  w3a 1084   = wceq 1538  wnf 1785  wcel 2115  wral 3126  wrex 3127  Vcvv 3471  wss 3910   class class class wbr 5039  cmpt 5119  ran crn 5529  infcinf 8881  cr 10513  -∞cmnf 10650  *cxr 10651   < clt 10652  cle 10653
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793  ax-sep 5176  ax-nul 5183  ax-pow 5239  ax-pr 5303  ax-un 7436  ax-cnex 10570  ax-resscn 10571  ax-1cn 10572  ax-icn 10573  ax-addcl 10574  ax-addrcl 10575  ax-mulcl 10576  ax-mulrcl 10577  ax-mulcom 10578  ax-addass 10579  ax-mulass 10580  ax-distr 10581  ax-i2m1 10582  ax-1ne0 10583  ax-1rid 10584  ax-rnegex 10585  ax-rrecex 10586  ax-cnre 10587  ax-pre-lttri 10588  ax-pre-lttrn 10589  ax-pre-ltadd 10590  ax-pre-mulgt0 10591  ax-pre-sup 10592
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2623  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ne 3008  df-nel 3112  df-ral 3131  df-rex 3132  df-reu 3133  df-rmo 3134  df-rab 3135  df-v 3473  df-sbc 3750  df-csb 3858  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4267  df-if 4441  df-pw 4514  df-sn 4541  df-pr 4543  df-op 4547  df-uni 4812  df-br 5040  df-opab 5102  df-mpt 5120  df-id 5433  df-po 5447  df-so 5448  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-riota 7088  df-ov 7133  df-oprab 7134  df-mpo 7135  df-er 8264  df-en 8485  df-dom 8486  df-sdom 8487  df-sup 8882  df-inf 8883  df-pnf 10654  df-mnf 10655  df-xr 10656  df-ltxr 10657  df-le 10658  df-sub 10849  df-neg 10850
This theorem is referenced by:  limsupmnflem  42153
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