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Theorem pimgtmnf2 41245
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 3720 . . 3 {𝑥𝐴 ∣ -∞ < (𝐹𝑥)} ⊆ 𝐴
21a1i 11 . 2 (𝜑 → {𝑥𝐴 ∣ -∞ < (𝐹𝑥)} ⊆ 𝐴)
3 ssid 3657 . . . . 5 𝐴𝐴
43a1i 11 . . . 4 (𝜑𝐴𝐴)
5 pimgtmnf2.2 . . . . . . . 8 (𝜑𝐹:𝐴⟶ℝ)
65ffvelrnda 6399 . . . . . . 7 ((𝜑𝑦𝐴) → (𝐹𝑦) ∈ ℝ)
76mnfltd 11996 . . . . . 6 ((𝜑𝑦𝐴) → -∞ < (𝐹𝑦))
87ralrimiva 2995 . . . . 5 (𝜑 → ∀𝑦𝐴 -∞ < (𝐹𝑦))
9 nfcv 2793 . . . . . . 7 𝑥-∞
10 nfcv 2793 . . . . . . 7 𝑥 <
11 pimgtmnf2.1 . . . . . . . 8 𝑥𝐹
12 nfcv 2793 . . . . . . . 8 𝑥𝑦
1311, 12nffv 6236 . . . . . . 7 𝑥(𝐹𝑦)
149, 10, 13nfbr 4732 . . . . . 6 𝑥-∞ < (𝐹𝑦)
15 nfv 1883 . . . . . 6 𝑦-∞ < (𝐹𝑥)
16 fveq2 6229 . . . . . . 7 (𝑦 = 𝑥 → (𝐹𝑦) = (𝐹𝑥))
1716breq2d 4697 . . . . . 6 (𝑦 = 𝑥 → (-∞ < (𝐹𝑦) ↔ -∞ < (𝐹𝑥)))
1814, 15, 17cbvral 3197 . . . . 5 (∀𝑦𝐴 -∞ < (𝐹𝑦) ↔ ∀𝑥𝐴 -∞ < (𝐹𝑥))
198, 18sylib 208 . . . 4 (𝜑 → ∀𝑥𝐴 -∞ < (𝐹𝑥))
204, 19jca 553 . . 3 (𝜑 → (𝐴𝐴 ∧ ∀𝑥𝐴 -∞ < (𝐹𝑥)))
21 nfcv 2793 . . . 4 𝑥𝐴
2221, 21ssrabf 39612 . . 3 (𝐴 ⊆ {𝑥𝐴 ∣ -∞ < (𝐹𝑥)} ↔ (𝐴𝐴 ∧ ∀𝑥𝐴 -∞ < (𝐹𝑥)))
2320, 22sylibr 224 . 2 (𝜑𝐴 ⊆ {𝑥𝐴 ∣ -∞ < (𝐹𝑥)})
242, 23eqssd 3653 1 (𝜑 → {𝑥𝐴 ∣ -∞ < (𝐹𝑥)} = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1523  wcel 2030  wnfc 2780  wral 2941  {crab 2945  wss 3607   class class class wbr 4685  wf 5922  cfv 5926  cr 9973  -∞cmnf 10110   < clt 10112
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-cnex 10030
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-fv 5934  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117
This theorem is referenced by:  pimgtmnf  41253
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