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Theorem pimltpnf2 41440
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 2913 . . . 4 𝑥𝐴
2 nfcv 2913 . . . 4 𝑦𝐴
3 nfv 1995 . . . 4 𝑦(𝐹𝑥) < +∞
4 pimltpnf2.1 . . . . . 6 𝑥𝐹
5 nfcv 2913 . . . . . 6 𝑥𝑦
64, 5nffv 6341 . . . . 5 𝑥(𝐹𝑦)
7 nfcv 2913 . . . . 5 𝑥 <
8 nfcv 2913 . . . . 5 𝑥+∞
96, 7, 8nfbr 4834 . . . 4 𝑥(𝐹𝑦) < +∞
10 fveq2 6333 . . . . 5 (𝑥 = 𝑦 → (𝐹𝑥) = (𝐹𝑦))
1110breq1d 4797 . . . 4 (𝑥 = 𝑦 → ((𝐹𝑥) < +∞ ↔ (𝐹𝑦) < +∞))
121, 2, 3, 9, 11cbvrab 3348 . . 3 {𝑥𝐴 ∣ (𝐹𝑥) < +∞} = {𝑦𝐴 ∣ (𝐹𝑦) < +∞}
1312a1i 11 . 2 (𝜑 → {𝑥𝐴 ∣ (𝐹𝑥) < +∞} = {𝑦𝐴 ∣ (𝐹𝑦) < +∞})
14 nfv 1995 . . 3 𝑦𝜑
15 pimltpnf2.2 . . . 4 (𝜑𝐹:𝐴⟶ℝ)
1615ffvelrnda 6504 . . 3 ((𝜑𝑦𝐴) → (𝐹𝑦) ∈ ℝ)
1714, 16pimltpnf 41433 . 2 (𝜑 → {𝑦𝐴 ∣ (𝐹𝑦) < +∞} = 𝐴)
1813, 17eqtrd 2805 1 (𝜑 → {𝑥𝐴 ∣ (𝐹𝑥) < +∞} = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1631  wnfc 2900  {crab 3065   class class class wbr 4787  wf 6026  cfv 6030  cr 10141  +∞cpnf 10277   < clt 10280
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035  ax-un 7100  ax-cnex 10198
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-br 4788  df-opab 4848  df-id 5158  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-iota 5993  df-fun 6032  df-fn 6033  df-f 6034  df-fv 6038  df-pnf 10282  df-xr 10284  df-ltxr 10285
This theorem is referenced by:  smfpimltxr  41473
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