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Theorem pimltmnf2 45404
Description: Given a real-valued function, the preimage of an open interval, unbounded below, with upper bound -∞, is the empty set. (Contributed by Glauco Siliprandi, 26-Jun-2021.) (Revised by Glauco Siliprandi, 15-Dec-2024.)
Hypotheses
Ref Expression
pimltmnf2.1 β„²π‘₯𝐹
pimltmnf2.2 (πœ‘ β†’ 𝐹:π΄βŸΆβ„)
Assertion
Ref Expression
pimltmnf2 (πœ‘ β†’ {π‘₯ ∈ 𝐴 ∣ (πΉβ€˜π‘₯) < -∞} = βˆ…)
Distinct variable group:   π‘₯,𝐴
Allowed substitution hints:   πœ‘(π‘₯)   𝐹(π‘₯)

Proof of Theorem pimltmnf2
StepHypRef Expression
1 pimltmnf2.1 . 2 β„²π‘₯𝐹
2 nfcv 2903 . 2 β„²π‘₯𝐴
3 pimltmnf2.2 . 2 (πœ‘ β†’ 𝐹:π΄βŸΆβ„)
41, 2, 3pimltmnf2f 45403 1 (πœ‘ β†’ {π‘₯ ∈ 𝐴 ∣ (πΉβ€˜π‘₯) < -∞} = βˆ…)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   = wceq 1541  β„²wnfc 2883  {crab 3432  βˆ…c0 4322   class class class wbr 5148  βŸΆwf 6539  β€˜cfv 6543  β„cr 11108  -∞cmnf 11245   < clt 11247
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 2703  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7724  ax-cnex 11165  ax-resscn 11166
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 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-er 8702  df-en 8939  df-dom 8940  df-sdom 8941  df-pnf 11249  df-mnf 11250  df-xr 11251  df-ltxr 11252  df-le 11253
This theorem is referenced by: (None)
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