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Theorem pimgtmnf2 41437
Description: Given a real-valued function, the preimage of an open interval, unbounded above, with lower bound -∞, is the whole domain. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
Hypotheses
Ref Expression
pimgtmnf2.1 𝑥𝐹
pimgtmnf2.2 (𝜑𝐹:𝐴⟶ℝ)
Assertion
Ref Expression
pimgtmnf2 (𝜑 → {𝑥𝐴 ∣ -∞ < (𝐹𝑥)} = 𝐴)
Distinct variable group:   𝑥,𝐴
Allowed substitution hints:   𝜑(𝑥)   𝐹(𝑥)

Proof of Theorem pimgtmnf2
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 ssrab2 3836 . . 3 {𝑥𝐴 ∣ -∞ < (𝐹𝑥)} ⊆ 𝐴
21a1i 11 . 2 (𝜑 → {𝑥𝐴 ∣ -∞ < (𝐹𝑥)} ⊆ 𝐴)
3 ssid 3773 . . . . 5 𝐴𝐴
43a1i 11 . . . 4 (𝜑𝐴𝐴)
5 pimgtmnf2.2 . . . . . . . 8 (𝜑𝐹:𝐴⟶ℝ)
65ffvelrnda 6500 . . . . . . 7 ((𝜑𝑦𝐴) → (𝐹𝑦) ∈ ℝ)
76mnfltd 12156 . . . . . 6 ((𝜑𝑦𝐴) → -∞ < (𝐹𝑦))
87ralrimiva 3115 . . . . 5 (𝜑 → ∀𝑦𝐴 -∞ < (𝐹𝑦))
9 nfcv 2913 . . . . . . 7 𝑥-∞
10 nfcv 2913 . . . . . . 7 𝑥 <
11 pimgtmnf2.1 . . . . . . . 8 𝑥𝐹
12 nfcv 2913 . . . . . . . 8 𝑥𝑦
1311, 12nffv 6337 . . . . . . 7 𝑥(𝐹𝑦)
149, 10, 13nfbr 4833 . . . . . 6 𝑥-∞ < (𝐹𝑦)
15 nfv 1995 . . . . . 6 𝑦-∞ < (𝐹𝑥)
16 fveq2 6330 . . . . . . 7 (𝑦 = 𝑥 → (𝐹𝑦) = (𝐹𝑥))
1716breq2d 4798 . . . . . 6 (𝑦 = 𝑥 → (-∞ < (𝐹𝑦) ↔ -∞ < (𝐹𝑥)))
1814, 15, 17cbvral 3316 . . . . 5 (∀𝑦𝐴 -∞ < (𝐹𝑦) ↔ ∀𝑥𝐴 -∞ < (𝐹𝑥))
198, 18sylib 208 . . . 4 (𝜑 → ∀𝑥𝐴 -∞ < (𝐹𝑥))
204, 19jca 501 . . 3 (𝜑 → (𝐴𝐴 ∧ ∀𝑥𝐴 -∞ < (𝐹𝑥)))
21 nfcv 2913 . . . 4 𝑥𝐴
2221, 21ssrabf 39812 . . 3 (𝐴 ⊆ {𝑥𝐴 ∣ -∞ < (𝐹𝑥)} ↔ (𝐴𝐴 ∧ ∀𝑥𝐴 -∞ < (𝐹𝑥)))
2320, 22sylibr 224 . 2 (𝜑𝐴 ⊆ {𝑥𝐴 ∣ -∞ < (𝐹𝑥)})
242, 23eqssd 3769 1 (𝜑 → {𝑥𝐴 ∣ -∞ < (𝐹𝑥)} = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1631  wcel 2145  wnfc 2900  wral 3061  {crab 3065  wss 3723   class class class wbr 4786  wf 6025  cfv 6029  cr 10135  -∞cmnf 10272   < clt 10274
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 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7094  ax-cnex 10192
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 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-br 4787  df-opab 4847  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-iota 5992  df-fun 6031  df-fn 6032  df-f 6033  df-fv 6037  df-pnf 10276  df-mnf 10277  df-xr 10278  df-ltxr 10279
This theorem is referenced by:  pimgtmnf  41445
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