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Theorem pimltpnf2f 45265
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, 15-Dec-2024.)
Hypotheses
Ref Expression
pimltpnf2f.1 𝑥𝐹
pimltpnf2f.2 𝑥𝐴
pimltpnf2f.3 (𝜑𝐹:𝐴⟶ℝ)
Assertion
Ref Expression
pimltpnf2f (𝜑 → {𝑥𝐴 ∣ (𝐹𝑥) < +∞} = 𝐴)

Proof of Theorem pimltpnf2f
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 pimltpnf2f.2 . . 3 𝑥𝐴
2 nfcv 2903 . . 3 𝑦𝐴
3 nfv 1917 . . 3 𝑦(𝐹𝑥) < +∞
4 pimltpnf2f.1 . . . . 5 𝑥𝐹
5 nfcv 2903 . . . . 5 𝑥𝑦
64, 5nffv 6889 . . . 4 𝑥(𝐹𝑦)
7 nfcv 2903 . . . 4 𝑥 <
8 nfcv 2903 . . . 4 𝑥+∞
96, 7, 8nfbr 5189 . . 3 𝑥(𝐹𝑦) < +∞
10 fveq2 6879 . . . 4 (𝑥 = 𝑦 → (𝐹𝑥) = (𝐹𝑦))
1110breq1d 5152 . . 3 (𝑥 = 𝑦 → ((𝐹𝑥) < +∞ ↔ (𝐹𝑦) < +∞))
121, 2, 3, 9, 11cbvrabw 3467 . 2 {𝑥𝐴 ∣ (𝐹𝑥) < +∞} = {𝑦𝐴 ∣ (𝐹𝑦) < +∞}
13 nfv 1917 . . 3 𝑦𝜑
14 pimltpnf2f.3 . . . 4 (𝜑𝐹:𝐴⟶ℝ)
1514ffvelcdmda 7072 . . 3 ((𝜑𝑦𝐴) → (𝐹𝑦) ∈ ℝ)
1613, 15pimltpnf 45257 . 2 (𝜑 → {𝑦𝐴 ∣ (𝐹𝑦) < +∞} = 𝐴)
1712, 16eqtrid 2784 1 (𝜑 → {𝑥𝐴 ∣ (𝐹𝑥) < +∞} = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1541  wnfc 2883  {crab 3432   class class class wbr 5142  wf 6529  cfv 6533  cr 11093  +∞cpnf 11229   < clt 11232
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 5293  ax-nul 5300  ax-pow 5357  ax-pr 5421  ax-un 7709  ax-cnex 11150
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-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-nul 4320  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5143  df-opab 5205  df-id 5568  df-xp 5676  df-rel 5677  df-cnv 5678  df-co 5679  df-dm 5680  df-rn 5681  df-iota 6485  df-fun 6535  df-fn 6536  df-f 6537  df-fv 6541  df-pnf 11234  df-xr 11236  df-ltxr 11237
This theorem is referenced by:  pimltpnf2  45266  smfpimltxr  45300
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