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Theorem pimgtpnf2 43783
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 2899 . . . 4 𝑥𝐴
2 nfcv 2899 . . . 4 𝑦𝐴
3 nfv 1921 . . . 4 𝑦+∞ < (𝐹𝑥)
4 nfcv 2899 . . . . 5 𝑥+∞
5 nfcv 2899 . . . . 5 𝑥 <
6 pimgtpnf2.1 . . . . . 6 𝑥𝐹
7 nfcv 2899 . . . . . 6 𝑥𝑦
86, 7nffv 6684 . . . . 5 𝑥(𝐹𝑦)
94, 5, 8nfbr 5077 . . . 4 𝑥+∞ < (𝐹𝑦)
10 fveq2 6674 . . . . 5 (𝑥 = 𝑦 → (𝐹𝑥) = (𝐹𝑦))
1110breq2d 5042 . . . 4 (𝑥 = 𝑦 → (+∞ < (𝐹𝑥) ↔ +∞ < (𝐹𝑦)))
121, 2, 3, 9, 11cbvrabw 3391 . . 3 {𝑥𝐴 ∣ +∞ < (𝐹𝑥)} = {𝑦𝐴 ∣ +∞ < (𝐹𝑦)}
1312a1i 11 . 2 (𝜑 → {𝑥𝐴 ∣ +∞ < (𝐹𝑥)} = {𝑦𝐴 ∣ +∞ < (𝐹𝑦)})
14 pimgtpnf2.2 . . . . . . . 8 (𝜑𝐹:𝐴⟶ℝ)
1514ffvelrnda 6861 . . . . . . 7 ((𝜑𝑦𝐴) → (𝐹𝑦) ∈ ℝ)
1615rexrd 10769 . . . . . 6 ((𝜑𝑦𝐴) → (𝐹𝑦) ∈ ℝ*)
17 pnfxr 10773 . . . . . . 7 +∞ ∈ ℝ*
1817a1i 11 . . . . . 6 ((𝜑𝑦𝐴) → +∞ ∈ ℝ*)
1915ltpnfd 12599 . . . . . 6 ((𝜑𝑦𝐴) → (𝐹𝑦) < +∞)
2016, 18, 19xrltled 12626 . . . . 5 ((𝜑𝑦𝐴) → (𝐹𝑦) ≤ +∞)
2116, 18xrlenltd 10785 . . . . 5 ((𝜑𝑦𝐴) → ((𝐹𝑦) ≤ +∞ ↔ ¬ +∞ < (𝐹𝑦)))
2220, 21mpbid 235 . . . 4 ((𝜑𝑦𝐴) → ¬ +∞ < (𝐹𝑦))
2322ralrimiva 3096 . . 3 (𝜑 → ∀𝑦𝐴 ¬ +∞ < (𝐹𝑦))
24 rabeq0 4273 . . 3 ({𝑦𝐴 ∣ +∞ < (𝐹𝑦)} = ∅ ↔ ∀𝑦𝐴 ¬ +∞ < (𝐹𝑦))
2523, 24sylibr 237 . 2 (𝜑 → {𝑦𝐴 ∣ +∞ < (𝐹𝑦)} = ∅)
2613, 25eqtrd 2773 1 (𝜑 → {𝑥𝐴 ∣ +∞ < (𝐹𝑥)} = ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 399   = wceq 1542  wcel 2114  wnfc 2879  wral 3053  {crab 3057  c0 4211   class class class wbr 5030  wf 6335  cfv 6339  cr 10614  +∞cpnf 10750  *cxr 10752   < clt 10753  cle 10754
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2710  ax-sep 5167  ax-nul 5174  ax-pow 5232  ax-pr 5296  ax-un 7479  ax-cnex 10671  ax-resscn 10672  ax-pre-lttri 10689  ax-pre-lttrn 10690
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-nel 3039  df-ral 3058  df-rex 3059  df-rab 3062  df-v 3400  df-sbc 3681  df-csb 3791  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-nul 4212  df-if 4415  df-pw 4490  df-sn 4517  df-pr 4519  df-op 4523  df-uni 4797  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5429  df-po 5442  df-so 5443  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-res 5537  df-ima 5538  df-iota 6297  df-fun 6341  df-fn 6342  df-f 6343  df-f1 6344  df-fo 6345  df-f1o 6346  df-fv 6347  df-er 8320  df-en 8556  df-dom 8557  df-sdom 8558  df-pnf 10755  df-mnf 10756  df-xr 10757  df-ltxr 10758  df-le 10759
This theorem is referenced by:  smfpimgtxr  43854
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