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Theorem infxrunb3rnmpt 40394
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 4940 . . . . . . 7 𝑥(𝑥𝐴𝐵)
43nfrn 5572 . . . . . 6 𝑥ran (𝑥𝐴𝐵)
5 nfv 2010 . . . . . 6 𝑥 𝑧𝑦
64, 5nfrex 3187 . . . . 5 𝑥𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝑦
7 simpr 478 . . . . . . . . 9 ((𝜑𝑥𝐴) → 𝑥𝐴)
8 infxrunb3rnmpt.3 . . . . . . . . 9 ((𝜑𝑥𝐴) → 𝐵 ∈ ℝ*)
9 eqid 2799 . . . . . . . . . 10 (𝑥𝐴𝐵) = (𝑥𝐴𝐵)
109elrnmpt1 5578 . . . . . . . . 9 ((𝑥𝐴𝐵 ∈ ℝ*) → 𝐵 ∈ ran (𝑥𝐴𝐵))
117, 8, 10syl2anc 580 . . . . . . . 8 ((𝜑𝑥𝐴) → 𝐵 ∈ ran (𝑥𝐴𝐵))
12113adant3 1163 . . . . . . 7 ((𝜑𝑥𝐴𝐵𝑦) → 𝐵 ∈ ran (𝑥𝐴𝐵))
13 simp3 1169 . . . . . . 7 ((𝜑𝑥𝐴𝐵𝑦) → 𝐵𝑦)
14 breq1 4846 . . . . . . . 8 (𝑧 = 𝐵 → (𝑧𝑦𝐵𝑦))
1514rspcev 3497 . . . . . . 7 ((𝐵 ∈ ran (𝑥𝐴𝐵) ∧ 𝐵𝑦) → ∃𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝑦)
1612, 13, 15syl2anc 580 . . . . . 6 ((𝜑𝑥𝐴𝐵𝑦) → ∃𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝑦)
17163exp 1149 . . . . 5 (𝜑 → (𝑥𝐴 → (𝐵𝑦 → ∃𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝑦)))
182, 6, 17rexlimd 3207 . . . 4 (𝜑 → (∃𝑥𝐴 𝐵𝑦 → ∃𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝑦))
19 nfv 2010 . . . . . 6 𝑧𝑥𝐴 𝐵𝑦
20 vex 3388 . . . . . . . . 9 𝑧 ∈ V
219elrnmpt 5576 . . . . . . . . 9 (𝑧 ∈ V → (𝑧 ∈ ran (𝑥𝐴𝐵) ↔ ∃𝑥𝐴 𝑧 = 𝐵))
2220, 21ax-mp 5 . . . . . . . 8 (𝑧 ∈ ran (𝑥𝐴𝐵) ↔ ∃𝑥𝐴 𝑧 = 𝐵)
2322biimpi 208 . . . . . . 7 (𝑧 ∈ ran (𝑥𝐴𝐵) → ∃𝑥𝐴 𝑧 = 𝐵)
2414biimpcd 241 . . . . . . . . . 10 (𝑧𝑦 → (𝑧 = 𝐵𝐵𝑦))
2524a1d 25 . . . . . . . . 9 (𝑧𝑦 → (𝑥𝐴 → (𝑧 = 𝐵𝐵𝑦)))
265, 25reximdai 3192 . . . . . . . 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 3205 . . . . 5 (∃𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝑦 → ∃𝑥𝐴 𝐵𝑦)
3029a1i 11 . . . 4 (𝜑 → (∃𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝑦 → ∃𝑥𝐴 𝐵𝑦))
3118, 30impbid 204 . . 3 (𝜑 → (∃𝑥𝐴 𝐵𝑦 ↔ ∃𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝑦))
321, 31ralbid 3164 . 2 (𝜑 → (∀𝑦 ∈ ℝ ∃𝑥𝐴 𝐵𝑦 ↔ ∀𝑦 ∈ ℝ ∃𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝑦))
332, 9, 8rnmptssd 40135 . . 3 (𝜑 → ran (𝑥𝐴𝐵) ⊆ ℝ*)
34 infxrunb3 40390 . . 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 271 1 (𝜑 → (∀𝑦 ∈ ℝ ∃𝑥𝐴 𝐵𝑦 ↔ inf(ran (𝑥𝐴𝐵), ℝ*, < ) = -∞))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 385  w3a 1108   = wceq 1653  wnf 1879  wcel 2157  wral 3089  wrex 3090  Vcvv 3385  wss 3769   class class class wbr 4843  cmpt 4922  ran crn 5313  infcinf 8589  cr 10223  -∞cmnf 10361  *cxr 10362   < clt 10363  cle 10364
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5097  ax-un 7183  ax-cnex 10280  ax-resscn 10281  ax-1cn 10282  ax-icn 10283  ax-addcl 10284  ax-addrcl 10285  ax-mulcl 10286  ax-mulrcl 10287  ax-mulcom 10288  ax-addass 10289  ax-mulass 10290  ax-distr 10291  ax-i2m1 10292  ax-1ne0 10293  ax-1rid 10294  ax-rnegex 10295  ax-rrecex 10296  ax-cnre 10297  ax-pre-lttri 10298  ax-pre-lttrn 10299  ax-pre-ltadd 10300  ax-pre-mulgt0 10301  ax-pre-sup 10302
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3or 1109  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ne 2972  df-nel 3075  df-ral 3094  df-rex 3095  df-reu 3096  df-rmo 3097  df-rab 3098  df-v 3387  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-pw 4351  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629  df-br 4844  df-opab 4906  df-mpt 4923  df-id 5220  df-po 5233  df-so 5234  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-rn 5323  df-res 5324  df-ima 5325  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-f1 6106  df-fo 6107  df-f1o 6108  df-fv 6109  df-riota 6839  df-ov 6881  df-oprab 6882  df-mpt2 6883  df-er 7982  df-en 8196  df-dom 8197  df-sdom 8198  df-sup 8590  df-inf 8591  df-pnf 10365  df-mnf 10366  df-xr 10367  df-ltxr 10368  df-le 10369  df-sub 10558  df-neg 10559
This theorem is referenced by:  limsupmnflem  40692
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