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Theorem pimltpnf2 43275
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.)
Hypotheses
Ref Expression
pimltpnf2.1 𝑥𝐹
pimltpnf2.2 (𝜑𝐹:𝐴⟶ℝ)
Assertion
Ref Expression
pimltpnf2 (𝜑 → {𝑥𝐴 ∣ (𝐹𝑥) < +∞} = 𝐴)
Distinct variable group:   𝑥,𝐴
Allowed substitution hints:   𝜑(𝑥)   𝐹(𝑥)

Proof of Theorem pimltpnf2
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 nfcv 2982 . . . 4 𝑥𝐴
2 nfcv 2982 . . . 4 𝑦𝐴
3 nfv 1916 . . . 4 𝑦(𝐹𝑥) < +∞
4 pimltpnf2.1 . . . . . 6 𝑥𝐹
5 nfcv 2982 . . . . . 6 𝑥𝑦
64, 5nffv 6672 . . . . 5 𝑥(𝐹𝑦)
7 nfcv 2982 . . . . 5 𝑥 <
8 nfcv 2982 . . . . 5 𝑥+∞
96, 7, 8nfbr 5100 . . . 4 𝑥(𝐹𝑦) < +∞
10 fveq2 6662 . . . . 5 (𝑥 = 𝑦 → (𝐹𝑥) = (𝐹𝑦))
1110breq1d 5063 . . . 4 (𝑥 = 𝑦 → ((𝐹𝑥) < +∞ ↔ (𝐹𝑦) < +∞))
121, 2, 3, 9, 11cbvrabw 3476 . . 3 {𝑥𝐴 ∣ (𝐹𝑥) < +∞} = {𝑦𝐴 ∣ (𝐹𝑦) < +∞}
1312a1i 11 . 2 (𝜑 → {𝑥𝐴 ∣ (𝐹𝑥) < +∞} = {𝑦𝐴 ∣ (𝐹𝑦) < +∞})
14 nfv 1916 . . 3 𝑦𝜑
15 pimltpnf2.2 . . . 4 (𝜑𝐹:𝐴⟶ℝ)
1615ffvelrnda 6843 . . 3 ((𝜑𝑦𝐴) → (𝐹𝑦) ∈ ℝ)
1714, 16pimltpnf 43268 . 2 (𝜑 → {𝑦𝐴 ∣ (𝐹𝑦) < +∞} = 𝐴)
1813, 17eqtrd 2859 1 (𝜑 → {𝑥𝐴 ∣ (𝐹𝑥) < +∞} = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1538  wnfc 2962  {crab 3137   class class class wbr 5053  wf 6340  cfv 6344  cr 10535  +∞cpnf 10671   < clt 10674
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 2179  ax-ext 2796  ax-sep 5190  ax-nul 5197  ax-pow 5254  ax-pr 5318  ax-un 7456  ax-cnex 10592
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3483  df-sbc 3760  df-dif 3923  df-un 3925  df-in 3927  df-ss 3937  df-nul 4278  df-if 4452  df-pw 4525  df-sn 4552  df-pr 4554  df-op 4558  df-uni 4826  df-br 5054  df-opab 5116  df-id 5448  df-xp 5549  df-rel 5550  df-cnv 5551  df-co 5552  df-dm 5553  df-rn 5554  df-iota 6303  df-fun 6346  df-fn 6347  df-f 6348  df-fv 6352  df-pnf 10676  df-xr 10678  df-ltxr 10679
This theorem is referenced by:  smfpimltxr  43308
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