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Theorem pimgtmnf2 42429
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 3946 . . 3 {𝑥𝐴 ∣ -∞ < (𝐹𝑥)} ⊆ 𝐴
21a1i 11 . 2 (𝜑 → {𝑥𝐴 ∣ -∞ < (𝐹𝑥)} ⊆ 𝐴)
3 ssid 3879 . . . . 5 𝐴𝐴
43a1i 11 . . . 4 (𝜑𝐴𝐴)
5 pimgtmnf2.2 . . . . . . . 8 (𝜑𝐹:𝐴⟶ℝ)
65ffvelrnda 6676 . . . . . . 7 ((𝜑𝑦𝐴) → (𝐹𝑦) ∈ ℝ)
76mnfltd 12336 . . . . . 6 ((𝜑𝑦𝐴) → -∞ < (𝐹𝑦))
87ralrimiva 3132 . . . . 5 (𝜑 → ∀𝑦𝐴 -∞ < (𝐹𝑦))
9 nfcv 2932 . . . . . . 7 𝑥-∞
10 nfcv 2932 . . . . . . 7 𝑥 <
11 pimgtmnf2.1 . . . . . . . 8 𝑥𝐹
12 nfcv 2932 . . . . . . . 8 𝑥𝑦
1311, 12nffv 6509 . . . . . . 7 𝑥(𝐹𝑦)
149, 10, 13nfbr 4976 . . . . . 6 𝑥-∞ < (𝐹𝑦)
15 nfv 1873 . . . . . 6 𝑦-∞ < (𝐹𝑥)
16 fveq2 6499 . . . . . . 7 (𝑦 = 𝑥 → (𝐹𝑦) = (𝐹𝑥))
1716breq2d 4941 . . . . . 6 (𝑦 = 𝑥 → (-∞ < (𝐹𝑦) ↔ -∞ < (𝐹𝑥)))
1814, 15, 17cbvral 3379 . . . . 5 (∀𝑦𝐴 -∞ < (𝐹𝑦) ↔ ∀𝑥𝐴 -∞ < (𝐹𝑥))
198, 18sylib 210 . . . 4 (𝜑 → ∀𝑥𝐴 -∞ < (𝐹𝑥))
204, 19jca 504 . . 3 (𝜑 → (𝐴𝐴 ∧ ∀𝑥𝐴 -∞ < (𝐹𝑥)))
21 nfcv 2932 . . . 4 𝑥𝐴
2221, 21ssrabf 40810 . . 3 (𝐴 ⊆ {𝑥𝐴 ∣ -∞ < (𝐹𝑥)} ↔ (𝐴𝐴 ∧ ∀𝑥𝐴 -∞ < (𝐹𝑥)))
2320, 22sylibr 226 . 2 (𝜑𝐴 ⊆ {𝑥𝐴 ∣ -∞ < (𝐹𝑥)})
242, 23eqssd 3875 1 (𝜑 → {𝑥𝐴 ∣ -∞ < (𝐹𝑥)} = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 387   = wceq 1507  wcel 2050  wnfc 2916  wral 3088  {crab 3092  wss 3829   class class class wbr 4929  wf 6184  cfv 6188  cr 10334  -∞cmnf 10472   < clt 10474
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2750  ax-sep 5060  ax-nul 5067  ax-pow 5119  ax-pr 5186  ax-un 7279  ax-cnex 10391
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2759  df-cleq 2771  df-clel 2846  df-nfc 2918  df-ral 3093  df-rex 3094  df-rab 3097  df-v 3417  df-sbc 3682  df-dif 3832  df-un 3834  df-in 3836  df-ss 3843  df-nul 4179  df-if 4351  df-pw 4424  df-sn 4442  df-pr 4444  df-op 4448  df-uni 4713  df-br 4930  df-opab 4992  df-id 5312  df-xp 5413  df-rel 5414  df-cnv 5415  df-co 5416  df-dm 5417  df-rn 5418  df-iota 6152  df-fun 6190  df-fn 6191  df-f 6192  df-fv 6196  df-pnf 10476  df-mnf 10477  df-xr 10478  df-ltxr 10479
This theorem is referenced by:  pimgtmnf  42437
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