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Theorem infxrunb3rnmpt 40171
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 4899 . . . . . . 7 𝑥(𝑥𝐴𝐵)
43nfrn 5523 . . . . . 6 𝑥ran (𝑥𝐴𝐵)
5 nfv 1992 . . . . . 6 𝑥 𝑧𝑦
64, 5nfrex 3145 . . . . 5 𝑥𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝑦
7 simpr 479 . . . . . . . . 9 ((𝜑𝑥𝐴) → 𝑥𝐴)
8 infxrunb3rnmpt.3 . . . . . . . . 9 ((𝜑𝑥𝐴) → 𝐵 ∈ ℝ*)
9 eqid 2760 . . . . . . . . . 10 (𝑥𝐴𝐵) = (𝑥𝐴𝐵)
109elrnmpt1 5529 . . . . . . . . 9 ((𝑥𝐴𝐵 ∈ ℝ*) → 𝐵 ∈ ran (𝑥𝐴𝐵))
117, 8, 10syl2anc 696 . . . . . . . 8 ((𝜑𝑥𝐴) → 𝐵 ∈ ran (𝑥𝐴𝐵))
12113adant3 1127 . . . . . . 7 ((𝜑𝑥𝐴𝐵𝑦) → 𝐵 ∈ ran (𝑥𝐴𝐵))
13 simp3 1133 . . . . . . 7 ((𝜑𝑥𝐴𝐵𝑦) → 𝐵𝑦)
14 breq1 4807 . . . . . . . 8 (𝑧 = 𝐵 → (𝑧𝑦𝐵𝑦))
1514rspcev 3449 . . . . . . 7 ((𝐵 ∈ ran (𝑥𝐴𝐵) ∧ 𝐵𝑦) → ∃𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝑦)
1612, 13, 15syl2anc 696 . . . . . 6 ((𝜑𝑥𝐴𝐵𝑦) → ∃𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝑦)
17163exp 1113 . . . . 5 (𝜑 → (𝑥𝐴 → (𝐵𝑦 → ∃𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝑦)))
182, 6, 17rexlimd 3164 . . . 4 (𝜑 → (∃𝑥𝐴 𝐵𝑦 → ∃𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝑦))
19 nfv 1992 . . . . . 6 𝑧𝑥𝐴 𝐵𝑦
20 vex 3343 . . . . . . . . 9 𝑧 ∈ V
219elrnmpt 5527 . . . . . . . . 9 (𝑧 ∈ V → (𝑧 ∈ ran (𝑥𝐴𝐵) ↔ ∃𝑥𝐴 𝑧 = 𝐵))
2220, 21ax-mp 5 . . . . . . . 8 (𝑧 ∈ ran (𝑥𝐴𝐵) ↔ ∃𝑥𝐴 𝑧 = 𝐵)
2322biimpi 206 . . . . . . 7 (𝑧 ∈ ran (𝑥𝐴𝐵) → ∃𝑥𝐴 𝑧 = 𝐵)
2414biimpcd 239 . . . . . . . . . 10 (𝑧𝑦 → (𝑧 = 𝐵𝐵𝑦))
2524a1d 25 . . . . . . . . 9 (𝑧𝑦 → (𝑥𝐴 → (𝑧 = 𝐵𝐵𝑦)))
265, 25reximdai 3150 . . . . . . . 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 3162 . . . . 5 (∃𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝑦 → ∃𝑥𝐴 𝐵𝑦)
3029a1i 11 . . . 4 (𝜑 → (∃𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝑦 → ∃𝑥𝐴 𝐵𝑦))
3118, 30impbid 202 . . 3 (𝜑 → (∃𝑥𝐴 𝐵𝑦 ↔ ∃𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝑦))
321, 31ralbid 3121 . 2 (𝜑 → (∀𝑦 ∈ ℝ ∃𝑥𝐴 𝐵𝑦 ↔ ∀𝑦 ∈ ℝ ∃𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝑦))
332, 9, 8rnmptssd 39902 . . 3 (𝜑 → ran (𝑥𝐴𝐵) ⊆ ℝ*)
34 infxrunb3 40167 . . 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 268 1 (𝜑 → (∀𝑦 ∈ ℝ ∃𝑥𝐴 𝐵𝑦 ↔ inf(ran (𝑥𝐴𝐵), ℝ*, < ) = -∞))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3a 1072   = wceq 1632  wnf 1857  wcel 2139  wral 3050  wrex 3051  Vcvv 3340  wss 3715   class class class wbr 4804  cmpt 4881  ran crn 5267  infcinf 8514  cr 10147  -∞cmnf 10284  *cxr 10285   < clt 10286  cle 10287
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115  ax-cnex 10204  ax-resscn 10205  ax-1cn 10206  ax-icn 10207  ax-addcl 10208  ax-addrcl 10209  ax-mulcl 10210  ax-mulrcl 10211  ax-mulcom 10212  ax-addass 10213  ax-mulass 10214  ax-distr 10215  ax-i2m1 10216  ax-1ne0 10217  ax-1rid 10218  ax-rnegex 10219  ax-rrecex 10220  ax-cnre 10221  ax-pre-lttri 10222  ax-pre-lttrn 10223  ax-pre-ltadd 10224  ax-pre-mulgt0 10225  ax-pre-sup 10226
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-nel 3036  df-ral 3055  df-rex 3056  df-reu 3057  df-rmo 3058  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-po 5187  df-so 5188  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-riota 6775  df-ov 6817  df-oprab 6818  df-mpt2 6819  df-er 7913  df-en 8124  df-dom 8125  df-sdom 8126  df-sup 8515  df-inf 8516  df-pnf 10288  df-mnf 10289  df-xr 10290  df-ltxr 10291  df-le 10292  df-sub 10480  df-neg 10481
This theorem is referenced by:  limsupmnflem  40473
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