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Theorem pimgtmnf2 43336
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 4010 . . 3 {𝑥𝐴 ∣ -∞ < (𝐹𝑥)} ⊆ 𝐴
21a1i 11 . 2 (𝜑 → {𝑥𝐴 ∣ -∞ < (𝐹𝑥)} ⊆ 𝐴)
3 ssid 3940 . . . . 5 𝐴𝐴
43a1i 11 . . . 4 (𝜑𝐴𝐴)
5 pimgtmnf2.2 . . . . . . . 8 (𝜑𝐹:𝐴⟶ℝ)
65ffvelrnda 6832 . . . . . . 7 ((𝜑𝑦𝐴) → (𝐹𝑦) ∈ ℝ)
76mnfltd 12511 . . . . . 6 ((𝜑𝑦𝐴) → -∞ < (𝐹𝑦))
87ralrimiva 3152 . . . . 5 (𝜑 → ∀𝑦𝐴 -∞ < (𝐹𝑦))
9 nfcv 2958 . . . . . . 7 𝑥-∞
10 nfcv 2958 . . . . . . 7 𝑥 <
11 pimgtmnf2.1 . . . . . . . 8 𝑥𝐹
12 nfcv 2958 . . . . . . . 8 𝑥𝑦
1311, 12nffv 6659 . . . . . . 7 𝑥(𝐹𝑦)
149, 10, 13nfbr 5080 . . . . . 6 𝑥-∞ < (𝐹𝑦)
15 nfv 1915 . . . . . 6 𝑦-∞ < (𝐹𝑥)
16 fveq2 6649 . . . . . . 7 (𝑦 = 𝑥 → (𝐹𝑦) = (𝐹𝑥))
1716breq2d 5045 . . . . . 6 (𝑦 = 𝑥 → (-∞ < (𝐹𝑦) ↔ -∞ < (𝐹𝑥)))
1814, 15, 17cbvralw 3390 . . . . 5 (∀𝑦𝐴 -∞ < (𝐹𝑦) ↔ ∀𝑥𝐴 -∞ < (𝐹𝑥))
198, 18sylib 221 . . . 4 (𝜑 → ∀𝑥𝐴 -∞ < (𝐹𝑥))
204, 19jca 515 . . 3 (𝜑 → (𝐴𝐴 ∧ ∀𝑥𝐴 -∞ < (𝐹𝑥)))
21 nfcv 2958 . . . 4 𝑥𝐴
2221, 21ssrabf 41737 . . 3 (𝐴 ⊆ {𝑥𝐴 ∣ -∞ < (𝐹𝑥)} ↔ (𝐴𝐴 ∧ ∀𝑥𝐴 -∞ < (𝐹𝑥)))
2320, 22sylibr 237 . 2 (𝜑𝐴 ⊆ {𝑥𝐴 ∣ -∞ < (𝐹𝑥)})
242, 23eqssd 3935 1 (𝜑 → {𝑥𝐴 ∣ -∞ < (𝐹𝑥)} = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2112  wnfc 2939  wral 3109  {crab 3113  wss 3884   class class class wbr 5033  wf 6324  cfv 6328  cr 10529  -∞cmnf 10666   < clt 10668
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445  ax-cnex 10586
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-sbc 3724  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-br 5034  df-opab 5096  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-fv 6336  df-pnf 10670  df-mnf 10671  df-xr 10672  df-ltxr 10673
This theorem is referenced by:  pimgtmnf  43344
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