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Theorem pimltpnf2f 46633
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 2908 . . 3 𝑦𝐴
3 nfv 1913 . . 3 𝑦(𝐹𝑥) < +∞
4 pimltpnf2f.1 . . . . 5 𝑥𝐹
5 nfcv 2908 . . . . 5 𝑥𝑦
64, 5nffv 6930 . . . 4 𝑥(𝐹𝑦)
7 nfcv 2908 . . . 4 𝑥 <
8 nfcv 2908 . . . 4 𝑥+∞
96, 7, 8nfbr 5213 . . 3 𝑥(𝐹𝑦) < +∞
10 fveq2 6920 . . . 4 (𝑥 = 𝑦 → (𝐹𝑥) = (𝐹𝑦))
1110breq1d 5176 . . 3 (𝑥 = 𝑦 → ((𝐹𝑥) < +∞ ↔ (𝐹𝑦) < +∞))
121, 2, 3, 9, 11cbvrabw 3481 . 2 {𝑥𝐴 ∣ (𝐹𝑥) < +∞} = {𝑦𝐴 ∣ (𝐹𝑦) < +∞}
13 nfv 1913 . . 3 𝑦𝜑
14 pimltpnf2f.3 . . . 4 (𝜑𝐹:𝐴⟶ℝ)
1514ffvelcdmda 7118 . . 3 ((𝜑𝑦𝐴) → (𝐹𝑦) ∈ ℝ)
1613, 15pimltpnf 46625 . 2 (𝜑 → {𝑦𝐴 ∣ (𝐹𝑦) < +∞} = 𝐴)
1712, 16eqtrid 2792 1 (𝜑 → {𝑥𝐴 ∣ (𝐹𝑥) < +∞} = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537  wnfc 2893  {crab 3443   class class class wbr 5166  wf 6569  cfv 6573  cr 11183  +∞cpnf 11321   < clt 11324
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770  ax-cnex 11240
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-fv 6581  df-pnf 11326  df-xr 11328  df-ltxr 11329
This theorem is referenced by:  pimltpnf2  46634  smfpimltxr  46668
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