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Theorem pimgtpnf2f 46661
Description: Given a real-valued function, the preimage of an open interval, unbounded above, with lower bound +∞, is the empty set. (Contributed by Glauco Siliprandi, 15-Dec-2021.)
Hypotheses
Ref Expression
pimgtpnf2f.1 𝑥𝐹
pimgtpnf2f.2 𝑥𝐴
pimgtpnf2f.3 (𝜑𝐹:𝐴⟶ℝ)
Assertion
Ref Expression
pimgtpnf2f (𝜑 → {𝑥𝐴 ∣ +∞ < (𝐹𝑥)} = ∅)

Proof of Theorem pimgtpnf2f
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 pimgtpnf2f.2 . . 3 𝑥𝐴
2 nfcv 2903 . . 3 𝑦𝐴
3 nfv 1912 . . 3 𝑦+∞ < (𝐹𝑥)
4 nfcv 2903 . . . 4 𝑥+∞
5 nfcv 2903 . . . 4 𝑥 <
6 pimgtpnf2f.1 . . . . 5 𝑥𝐹
7 nfcv 2903 . . . . 5 𝑥𝑦
86, 7nffv 6917 . . . 4 𝑥(𝐹𝑦)
94, 5, 8nfbr 5195 . . 3 𝑥+∞ < (𝐹𝑦)
10 fveq2 6907 . . . 4 (𝑥 = 𝑦 → (𝐹𝑥) = (𝐹𝑦))
1110breq2d 5160 . . 3 (𝑥 = 𝑦 → (+∞ < (𝐹𝑥) ↔ +∞ < (𝐹𝑦)))
121, 2, 3, 9, 11cbvrabw 3471 . 2 {𝑥𝐴 ∣ +∞ < (𝐹𝑥)} = {𝑦𝐴 ∣ +∞ < (𝐹𝑦)}
13 pimgtpnf2f.3 . . . . . . . 8 (𝜑𝐹:𝐴⟶ℝ)
1413ffvelcdmda 7104 . . . . . . 7 ((𝜑𝑦𝐴) → (𝐹𝑦) ∈ ℝ)
1514rexrd 11309 . . . . . 6 ((𝜑𝑦𝐴) → (𝐹𝑦) ∈ ℝ*)
1615pnfged 45424 . . . . 5 ((𝜑𝑦𝐴) → (𝐹𝑦) ≤ +∞)
17 pnfxr 11313 . . . . . . 7 +∞ ∈ ℝ*
1817a1i 11 . . . . . 6 ((𝜑𝑦𝐴) → +∞ ∈ ℝ*)
1915, 18xrlenltd 11325 . . . . 5 ((𝜑𝑦𝐴) → ((𝐹𝑦) ≤ +∞ ↔ ¬ +∞ < (𝐹𝑦)))
2016, 19mpbid 232 . . . 4 ((𝜑𝑦𝐴) → ¬ +∞ < (𝐹𝑦))
2120ralrimiva 3144 . . 3 (𝜑 → ∀𝑦𝐴 ¬ +∞ < (𝐹𝑦))
22 rabeq0 4394 . . 3 ({𝑦𝐴 ∣ +∞ < (𝐹𝑦)} = ∅ ↔ ∀𝑦𝐴 ¬ +∞ < (𝐹𝑦))
2321, 22sylibr 234 . 2 (𝜑 → {𝑦𝐴 ∣ +∞ < (𝐹𝑦)} = ∅)
2412, 23eqtrid 2787 1 (𝜑 → {𝑥𝐴 ∣ +∞ < (𝐹𝑥)} = ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395   = wceq 1537  wcel 2106  wnfc 2888  wral 3059  {crab 3433  c0 4339   class class class wbr 5148  wf 6559  cfv 6563  cr 11152  +∞cpnf 11290  *cxr 11292   < clt 11293  cle 11294
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fv 6571  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299
This theorem is referenced by:  pimgtpnf2  46662  smfpimgtxr  46736
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