Metamath Proof Explorer


Theorem pimltmnf2

Description: Given a real-valued function, the preimage of an open interval, unbounded below, with upper bound -oo , is the empty set. (Contributed by Glauco Siliprandi, 26-Jun-2021)

Ref Expression
Hypotheses pimltmnf2.1 𝑥 𝐹
pimltmnf2.2 ( 𝜑𝐹 : 𝐴 ⟶ ℝ )
Assertion pimltmnf2 ( 𝜑 → { 𝑥𝐴 ∣ ( 𝐹𝑥 ) < -∞ } = ∅ )

Proof

Step Hyp Ref Expression
1 pimltmnf2.1 𝑥 𝐹
2 pimltmnf2.2 ( 𝜑𝐹 : 𝐴 ⟶ ℝ )
3 nfcv 𝑥 𝐴
4 nfcv 𝑦 𝐴
5 nfv 𝑦 ( 𝐹𝑥 ) < -∞
6 nfcv 𝑥 𝑦
7 1 6 nffv 𝑥 ( 𝐹𝑦 )
8 nfcv 𝑥 <
9 nfcv 𝑥 -∞
10 7 8 9 nfbr 𝑥 ( 𝐹𝑦 ) < -∞
11 fveq2 ( 𝑥 = 𝑦 → ( 𝐹𝑥 ) = ( 𝐹𝑦 ) )
12 11 breq1d ( 𝑥 = 𝑦 → ( ( 𝐹𝑥 ) < -∞ ↔ ( 𝐹𝑦 ) < -∞ ) )
13 3 4 5 10 12 cbvrabw { 𝑥𝐴 ∣ ( 𝐹𝑥 ) < -∞ } = { 𝑦𝐴 ∣ ( 𝐹𝑦 ) < -∞ }
14 13 a1i ( 𝜑 → { 𝑥𝐴 ∣ ( 𝐹𝑥 ) < -∞ } = { 𝑦𝐴 ∣ ( 𝐹𝑦 ) < -∞ } )
15 mnfxr -∞ ∈ ℝ*
16 15 a1i ( ( 𝜑𝑦𝐴 ) → -∞ ∈ ℝ* )
17 2 ffvelrnda ( ( 𝜑𝑦𝐴 ) → ( 𝐹𝑦 ) ∈ ℝ )
18 17 rexrd ( ( 𝜑𝑦𝐴 ) → ( 𝐹𝑦 ) ∈ ℝ* )
19 17 mnfltd ( ( 𝜑𝑦𝐴 ) → -∞ < ( 𝐹𝑦 ) )
20 16 18 19 xrltled ( ( 𝜑𝑦𝐴 ) → -∞ ≤ ( 𝐹𝑦 ) )
21 16 18 xrlenltd ( ( 𝜑𝑦𝐴 ) → ( -∞ ≤ ( 𝐹𝑦 ) ↔ ¬ ( 𝐹𝑦 ) < -∞ ) )
22 20 21 mpbid ( ( 𝜑𝑦𝐴 ) → ¬ ( 𝐹𝑦 ) < -∞ )
23 22 ralrimiva ( 𝜑 → ∀ 𝑦𝐴 ¬ ( 𝐹𝑦 ) < -∞ )
24 rabeq0 ( { 𝑦𝐴 ∣ ( 𝐹𝑦 ) < -∞ } = ∅ ↔ ∀ 𝑦𝐴 ¬ ( 𝐹𝑦 ) < -∞ )
25 23 24 sylibr ( 𝜑 → { 𝑦𝐴 ∣ ( 𝐹𝑦 ) < -∞ } = ∅ )
26 14 25 eqtrd ( 𝜑 → { 𝑥𝐴 ∣ ( 𝐹𝑥 ) < -∞ } = ∅ )