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Theorem pimltpnf2 45429
Description: Given a real-valued function, the preimage of an open interval, unbounded below, with upper bound +∞, is the whole domain. (Contributed by Glauco Siliprandi, 26-Jun-2021.) (Revised by Glauco Siliprandi, 15-Dec-2024.)
Hypotheses
Ref Expression
pimltpnf2.1 β„²π‘₯𝐹
pimltpnf2.2 (πœ‘ β†’ 𝐹:π΄βŸΆβ„)
Assertion
Ref Expression
pimltpnf2 (πœ‘ β†’ {π‘₯ ∈ 𝐴 ∣ (πΉβ€˜π‘₯) < +∞} = 𝐴)
Distinct variable group:   π‘₯,𝐴
Allowed substitution hints:   πœ‘(π‘₯)   𝐹(π‘₯)

Proof of Theorem pimltpnf2
StepHypRef Expression
1 pimltpnf2.1 . 2 β„²π‘₯𝐹
2 nfcv 2904 . 2 β„²π‘₯𝐴
3 pimltpnf2.2 . 2 (πœ‘ β†’ 𝐹:π΄βŸΆβ„)
41, 2, 3pimltpnf2f 45428 1 (πœ‘ β†’ {π‘₯ ∈ 𝐴 ∣ (πΉβ€˜π‘₯) < +∞} = 𝐴)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   = wceq 1542  β„²wnfc 2884  {crab 3433   class class class wbr 5149  βŸΆwf 6540  β€˜cfv 6544  β„cr 11109  +∞cpnf 11245   < clt 11248
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725  ax-cnex 11166
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-fv 6552  df-pnf 11250  df-xr 11252  df-ltxr 11253
This theorem is referenced by: (None)
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