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Theorem pimltpnf2 39600
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 2751 . . . 4 𝑥𝐴
2 nfcv 2751 . . . 4 𝑦𝐴
3 nfv 1830 . . . 4 𝑦(𝐹𝑥) < +∞
4 pimltpnf2.1 . . . . . 6 𝑥𝐹
5 nfcv 2751 . . . . . 6 𝑥𝑦
64, 5nffv 6110 . . . . 5 𝑥(𝐹𝑦)
7 nfcv 2751 . . . . 5 𝑥 <
8 nfcv 2751 . . . . 5 𝑥+∞
96, 7, 8nfbr 4629 . . . 4 𝑥(𝐹𝑦) < +∞
10 fveq2 6103 . . . . 5 (𝑥 = 𝑦 → (𝐹𝑥) = (𝐹𝑦))
1110breq1d 4593 . . . 4 (𝑥 = 𝑦 → ((𝐹𝑥) < +∞ ↔ (𝐹𝑦) < +∞))
121, 2, 3, 9, 11cbvrab 3171 . . 3 {𝑥𝐴 ∣ (𝐹𝑥) < +∞} = {𝑦𝐴 ∣ (𝐹𝑦) < +∞}
1312a1i 11 . 2 (𝜑 → {𝑥𝐴 ∣ (𝐹𝑥) < +∞} = {𝑦𝐴 ∣ (𝐹𝑦) < +∞})
14 nfv 1830 . . 3 𝑦𝜑
15 pimltpnf2.2 . . . 4 (𝜑𝐹:𝐴⟶ℝ)
1615ffvelrnda 6267 . . 3 ((𝜑𝑦𝐴) → (𝐹𝑦) ∈ ℝ)
1714, 16pimltpnf 39593 . 2 (𝜑 → {𝑦𝐴 ∣ (𝐹𝑦) < +∞} = 𝐴)
1813, 17eqtrd 2644 1 (𝜑 → {𝑥𝐴 ∣ (𝐹𝑥) < +∞} = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1475  wnfc 2738  {crab 2900   class class class wbr 4583  wf 5800  cfv 5804  cr 9814  +∞cpnf 9950   < clt 9953
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-fv 5812  df-pnf 9955  df-xr 9957  df-ltxr 9958
This theorem is referenced by:  smfpimltxr  39634
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