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Theorem pimgtpnf2f 46720
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 2905 . . 3 𝑦𝐴
3 nfv 1914 . . 3 𝑦+∞ < (𝐹𝑥)
4 nfcv 2905 . . . 4 𝑥+∞
5 nfcv 2905 . . . 4 𝑥 <
6 pimgtpnf2f.1 . . . . 5 𝑥𝐹
7 nfcv 2905 . . . . 5 𝑥𝑦
86, 7nffv 6916 . . . 4 𝑥(𝐹𝑦)
94, 5, 8nfbr 5190 . . 3 𝑥+∞ < (𝐹𝑦)
10 fveq2 6906 . . . 4 (𝑥 = 𝑦 → (𝐹𝑥) = (𝐹𝑦))
1110breq2d 5155 . . 3 (𝑥 = 𝑦 → (+∞ < (𝐹𝑥) ↔ +∞ < (𝐹𝑦)))
121, 2, 3, 9, 11cbvrabw 3473 . 2 {𝑥𝐴 ∣ +∞ < (𝐹𝑥)} = {𝑦𝐴 ∣ +∞ < (𝐹𝑦)}
13 pimgtpnf2f.3 . . . . . . . 8 (𝜑𝐹:𝐴⟶ℝ)
1413ffvelcdmda 7104 . . . . . . 7 ((𝜑𝑦𝐴) → (𝐹𝑦) ∈ ℝ)
1514rexrd 11311 . . . . . 6 ((𝜑𝑦𝐴) → (𝐹𝑦) ∈ ℝ*)
1615pnfged 13173 . . . . 5 ((𝜑𝑦𝐴) → (𝐹𝑦) ≤ +∞)
17 pnfxr 11315 . . . . . . 7 +∞ ∈ ℝ*
1817a1i 11 . . . . . 6 ((𝜑𝑦𝐴) → +∞ ∈ ℝ*)
1915, 18xrlenltd 11327 . . . . 5 ((𝜑𝑦𝐴) → ((𝐹𝑦) ≤ +∞ ↔ ¬ +∞ < (𝐹𝑦)))
2016, 19mpbid 232 . . . 4 ((𝜑𝑦𝐴) → ¬ +∞ < (𝐹𝑦))
2120ralrimiva 3146 . . 3 (𝜑 → ∀𝑦𝐴 ¬ +∞ < (𝐹𝑦))
22 rabeq0 4388 . . 3 ({𝑦𝐴 ∣ +∞ < (𝐹𝑦)} = ∅ ↔ ∀𝑦𝐴 ¬ +∞ < (𝐹𝑦))
2321, 22sylibr 234 . 2 (𝜑 → {𝑦𝐴 ∣ +∞ < (𝐹𝑦)} = ∅)
2412, 23eqtrid 2789 1 (𝜑 → {𝑥𝐴 ∣ +∞ < (𝐹𝑥)} = ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395   = wceq 1540  wcel 2108  wnfc 2890  wral 3061  {crab 3436  c0 4333   class class class wbr 5143  wf 6557  cfv 6561  cr 11154  +∞cpnf 11292  *cxr 11294   < clt 11295  cle 11296
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-fv 6569  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301
This theorem is referenced by:  pimgtpnf2  46721  smfpimgtxr  46795
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