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Theorem pimgtpnf2 39594
 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, 26-Jun-2021.)
Hypotheses
Ref Expression
pimgtpnf2.1 𝑥𝐹
pimgtpnf2.2 (𝜑𝐹:𝐴⟶ℝ)
Assertion
Ref Expression
pimgtpnf2 (𝜑 → {𝑥𝐴 ∣ +∞ < (𝐹𝑥)} = ∅)
Distinct variable group:   𝑥,𝐴
Allowed substitution hints:   𝜑(𝑥)   𝐹(𝑥)

Proof of Theorem pimgtpnf2
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 nfcv 2751 . . . . 5 𝑥+∞
5 nfcv 2751 . . . . 5 𝑥 <
6 pimgtpnf2.1 . . . . . 6 𝑥𝐹
7 nfcv 2751 . . . . . 6 𝑥𝑦
86, 7nffv 6110 . . . . 5 𝑥(𝐹𝑦)
94, 5, 8nfbr 4629 . . . 4 𝑥+∞ < (𝐹𝑦)
10 fveq2 6103 . . . . 5 (𝑥 = 𝑦 → (𝐹𝑥) = (𝐹𝑦))
1110breq2d 4595 . . . 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 pimgtpnf2.2 . . . . . . . 8 (𝜑𝐹:𝐴⟶ℝ)
1514ffvelrnda 6267 . . . . . . 7 ((𝜑𝑦𝐴) → (𝐹𝑦) ∈ ℝ)
1615rexrd 9968 . . . . . 6 ((𝜑𝑦𝐴) → (𝐹𝑦) ∈ ℝ*)
17 pnfxr 9971 . . . . . . 7 +∞ ∈ ℝ*
1817a1i 11 . . . . . 6 ((𝜑𝑦𝐴) → +∞ ∈ ℝ*)
1915ltpnfd 11831 . . . . . 6 ((𝜑𝑦𝐴) → (𝐹𝑦) < +∞)
2016, 18, 19xrltled 38427 . . . . 5 ((𝜑𝑦𝐴) → (𝐹𝑦) ≤ +∞)
2116, 18xrlenltd 9983 . . . . 5 ((𝜑𝑦𝐴) → ((𝐹𝑦) ≤ +∞ ↔ ¬ +∞ < (𝐹𝑦)))
2220, 21mpbid 221 . . . 4 ((𝜑𝑦𝐴) → ¬ +∞ < (𝐹𝑦))
2322ralrimiva 2949 . . 3 (𝜑 → ∀𝑦𝐴 ¬ +∞ < (𝐹𝑦))
24 rabeq0 3911 . . 3 ({𝑦𝐴 ∣ +∞ < (𝐹𝑦)} = ∅ ↔ ∀𝑦𝐴 ¬ +∞ < (𝐹𝑦))
2523, 24sylibr 223 . 2 (𝜑 → {𝑦𝐴 ∣ +∞ < (𝐹𝑦)} = ∅)
2613, 25eqtrd 2644 1 (𝜑 → {𝑥𝐴 ∣ +∞ < (𝐹𝑥)} = ∅)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  Ⅎwnfc 2738  ∀wral 2896  {crab 2900  ∅c0 3874   class class class wbr 4583  ⟶wf 5800  ‘cfv 5804  ℝcr 9814  +∞cpnf 9950  ℝ*cxr 9952   < clt 9953   ≤ cle 9954 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  ax-resscn 9872  ax-pre-lttri 9889  ax-pre-lttrn 9890 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  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-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  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-mpt 4645  df-id 4953  df-po 4959  df-so 4960  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959 This theorem is referenced by:  smfpimgtxr  39666
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