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Theorem pimgtmnf2 46670
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 4090 . . 3 {𝑥𝐴 ∣ -∞ < (𝐹𝑥)} ⊆ 𝐴
21a1i 11 . 2 (𝜑 → {𝑥𝐴 ∣ -∞ < (𝐹𝑥)} ⊆ 𝐴)
3 ssid 4018 . . . . 5 𝐴𝐴
43a1i 11 . . . 4 (𝜑𝐴𝐴)
5 pimgtmnf2.2 . . . . . . . 8 (𝜑𝐹:𝐴⟶ℝ)
65ffvelcdmda 7104 . . . . . . 7 ((𝜑𝑦𝐴) → (𝐹𝑦) ∈ ℝ)
76mnfltd 13164 . . . . . 6 ((𝜑𝑦𝐴) → -∞ < (𝐹𝑦))
87ralrimiva 3144 . . . . 5 (𝜑 → ∀𝑦𝐴 -∞ < (𝐹𝑦))
9 nfcv 2903 . . . . . . 7 𝑥-∞
10 nfcv 2903 . . . . . . 7 𝑥 <
11 pimgtmnf2.1 . . . . . . . 8 𝑥𝐹
12 nfcv 2903 . . . . . . . 8 𝑥𝑦
1311, 12nffv 6917 . . . . . . 7 𝑥(𝐹𝑦)
149, 10, 13nfbr 5195 . . . . . 6 𝑥-∞ < (𝐹𝑦)
15 nfv 1912 . . . . . 6 𝑦-∞ < (𝐹𝑥)
16 fveq2 6907 . . . . . . 7 (𝑦 = 𝑥 → (𝐹𝑦) = (𝐹𝑥))
1716breq2d 5160 . . . . . 6 (𝑦 = 𝑥 → (-∞ < (𝐹𝑦) ↔ -∞ < (𝐹𝑥)))
1814, 15, 17cbvralw 3304 . . . . 5 (∀𝑦𝐴 -∞ < (𝐹𝑦) ↔ ∀𝑥𝐴 -∞ < (𝐹𝑥))
198, 18sylib 218 . . . 4 (𝜑 → ∀𝑥𝐴 -∞ < (𝐹𝑥))
204, 19jca 511 . . 3 (𝜑 → (𝐴𝐴 ∧ ∀𝑥𝐴 -∞ < (𝐹𝑥)))
21 nfcv 2903 . . . 4 𝑥𝐴
2221, 21ssrabf 45054 . . 3 (𝐴 ⊆ {𝑥𝐴 ∣ -∞ < (𝐹𝑥)} ↔ (𝐴𝐴 ∧ ∀𝑥𝐴 -∞ < (𝐹𝑥)))
2320, 22sylibr 234 . 2 (𝜑𝐴 ⊆ {𝑥𝐴 ∣ -∞ < (𝐹𝑥)})
242, 23eqssd 4013 1 (𝜑 → {𝑥𝐴 ∣ -∞ < (𝐹𝑥)} = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wcel 2106  wnfc 2888  wral 3059  {crab 3433  wss 3963   class class class wbr 5148  wf 6559  cfv 6563  cr 11152  -∞cmnf 11291   < clt 11293
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
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-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  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
This theorem is referenced by: (None)
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