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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > pimltmnf2f | Structured version Visualization version GIF version |
Description: Given a real-valued function, the preimage of an open interval, unbounded below, with upper bound -∞, is the empty set. (Contributed by Glauco Siliprandi, 15-Dec-2024.) |
Ref | Expression |
---|---|
pimltmnf2f.1 | ⊢ Ⅎ𝑥𝐹 |
pimltmnf2f.2 | ⊢ Ⅎ𝑥𝐴 |
pimltmnf2f.3 | ⊢ (𝜑 → 𝐹:𝐴⟶ℝ) |
Ref | Expression |
---|---|
pimltmnf2f | ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) < -∞} = ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pimltmnf2f.2 | . . 3 ⊢ Ⅎ𝑥𝐴 | |
2 | nfcv 2903 | . . 3 ⊢ Ⅎ𝑦𝐴 | |
3 | nfv 1917 | . . 3 ⊢ Ⅎ𝑦(𝐹‘𝑥) < -∞ | |
4 | pimltmnf2f.1 | . . . . 5 ⊢ Ⅎ𝑥𝐹 | |
5 | nfcv 2903 | . . . . 5 ⊢ Ⅎ𝑥𝑦 | |
6 | 4, 5 | nffv 6889 | . . . 4 ⊢ Ⅎ𝑥(𝐹‘𝑦) |
7 | nfcv 2903 | . . . 4 ⊢ Ⅎ𝑥 < | |
8 | nfcv 2903 | . . . 4 ⊢ Ⅎ𝑥-∞ | |
9 | 6, 7, 8 | nfbr 5189 | . . 3 ⊢ Ⅎ𝑥(𝐹‘𝑦) < -∞ |
10 | fveq2 6879 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝐹‘𝑥) = (𝐹‘𝑦)) | |
11 | 10 | breq1d 5152 | . . 3 ⊢ (𝑥 = 𝑦 → ((𝐹‘𝑥) < -∞ ↔ (𝐹‘𝑦) < -∞)) |
12 | 1, 2, 3, 9, 11 | cbvrabw 3467 | . 2 ⊢ {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) < -∞} = {𝑦 ∈ 𝐴 ∣ (𝐹‘𝑦) < -∞} |
13 | pimltmnf2f.3 | . . . . . . . 8 ⊢ (𝜑 → 𝐹:𝐴⟶ℝ) | |
14 | 13 | ffvelcdmda 7072 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → (𝐹‘𝑦) ∈ ℝ) |
15 | 14 | rexrd 11248 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → (𝐹‘𝑦) ∈ ℝ*) |
16 | 15 | mnfled 13099 | . . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → -∞ ≤ (𝐹‘𝑦)) |
17 | mnfxr 11255 | . . . . . . 7 ⊢ -∞ ∈ ℝ* | |
18 | 17 | a1i 11 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → -∞ ∈ ℝ*) |
19 | 18, 15 | xrlenltd 11264 | . . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → (-∞ ≤ (𝐹‘𝑦) ↔ ¬ (𝐹‘𝑦) < -∞)) |
20 | 16, 19 | mpbid 231 | . . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → ¬ (𝐹‘𝑦) < -∞) |
21 | 20 | ralrimiva 3146 | . . 3 ⊢ (𝜑 → ∀𝑦 ∈ 𝐴 ¬ (𝐹‘𝑦) < -∞) |
22 | rabeq0 4381 | . . 3 ⊢ ({𝑦 ∈ 𝐴 ∣ (𝐹‘𝑦) < -∞} = ∅ ↔ ∀𝑦 ∈ 𝐴 ¬ (𝐹‘𝑦) < -∞) | |
23 | 21, 22 | sylibr 233 | . 2 ⊢ (𝜑 → {𝑦 ∈ 𝐴 ∣ (𝐹‘𝑦) < -∞} = ∅) |
24 | 12, 23 | eqtrid 2784 | 1 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) < -∞} = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 Ⅎwnfc 2883 ∀wral 3061 {crab 3432 ∅c0 4319 class class class wbr 5142 ⟶wf 6529 ‘cfv 6533 ℝcr 11093 -∞cmnf 11230 ℝ*cxr 11231 < clt 11232 ≤ cle 11233 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5293 ax-nul 5300 ax-pow 5357 ax-pr 5421 ax-un 7709 ax-cnex 11150 ax-resscn 11151 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-sbc 3775 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-nul 4320 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-br 5143 df-opab 5205 df-mpt 5226 df-id 5568 df-xp 5676 df-rel 5677 df-cnv 5678 df-co 5679 df-dm 5680 df-rn 5681 df-res 5682 df-ima 5683 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-er 8688 df-en 8925 df-dom 8926 df-sdom 8927 df-pnf 11234 df-mnf 11235 df-xr 11236 df-ltxr 11237 df-le 11238 |
This theorem is referenced by: pimltmnf2 45251 smfpimltxr 45300 |
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