Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > pimltmnf2 | 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, 26-Jun-2021.) |
Ref | Expression |
---|---|
pimltmnf2.1 | ⊢ Ⅎ𝑥𝐹 |
pimltmnf2.2 | ⊢ (𝜑 → 𝐹:𝐴⟶ℝ) |
Ref | Expression |
---|---|
pimltmnf2 | ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) < -∞} = ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfcv 2900 | . . . 4 ⊢ Ⅎ𝑥𝐴 | |
2 | nfcv 2900 | . . . 4 ⊢ Ⅎ𝑦𝐴 | |
3 | nfv 1921 | . . . 4 ⊢ Ⅎ𝑦(𝐹‘𝑥) < -∞ | |
4 | pimltmnf2.1 | . . . . . 6 ⊢ Ⅎ𝑥𝐹 | |
5 | nfcv 2900 | . . . . . 6 ⊢ Ⅎ𝑥𝑦 | |
6 | 4, 5 | nffv 6687 | . . . . 5 ⊢ Ⅎ𝑥(𝐹‘𝑦) |
7 | nfcv 2900 | . . . . 5 ⊢ Ⅎ𝑥 < | |
8 | nfcv 2900 | . . . . 5 ⊢ Ⅎ𝑥-∞ | |
9 | 6, 7, 8 | nfbr 5078 | . . . 4 ⊢ Ⅎ𝑥(𝐹‘𝑦) < -∞ |
10 | fveq2 6677 | . . . . 5 ⊢ (𝑥 = 𝑦 → (𝐹‘𝑥) = (𝐹‘𝑦)) | |
11 | 10 | breq1d 5041 | . . . 4 ⊢ (𝑥 = 𝑦 → ((𝐹‘𝑥) < -∞ ↔ (𝐹‘𝑦) < -∞)) |
12 | 1, 2, 3, 9, 11 | cbvrabw 3392 | . . 3 ⊢ {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) < -∞} = {𝑦 ∈ 𝐴 ∣ (𝐹‘𝑦) < -∞} |
13 | 12 | a1i 11 | . 2 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) < -∞} = {𝑦 ∈ 𝐴 ∣ (𝐹‘𝑦) < -∞}) |
14 | mnfxr 10779 | . . . . . . 7 ⊢ -∞ ∈ ℝ* | |
15 | 14 | a1i 11 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → -∞ ∈ ℝ*) |
16 | pimltmnf2.2 | . . . . . . . 8 ⊢ (𝜑 → 𝐹:𝐴⟶ℝ) | |
17 | 16 | ffvelrnda 6864 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → (𝐹‘𝑦) ∈ ℝ) |
18 | 17 | rexrd 10772 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → (𝐹‘𝑦) ∈ ℝ*) |
19 | 17 | mnfltd 12605 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → -∞ < (𝐹‘𝑦)) |
20 | 15, 18, 19 | xrltled 12629 | . . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → -∞ ≤ (𝐹‘𝑦)) |
21 | 15, 18 | xrlenltd 10788 | . . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → (-∞ ≤ (𝐹‘𝑦) ↔ ¬ (𝐹‘𝑦) < -∞)) |
22 | 20, 21 | mpbid 235 | . . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → ¬ (𝐹‘𝑦) < -∞) |
23 | 22 | ralrimiva 3097 | . . 3 ⊢ (𝜑 → ∀𝑦 ∈ 𝐴 ¬ (𝐹‘𝑦) < -∞) |
24 | rabeq0 4274 | . . 3 ⊢ ({𝑦 ∈ 𝐴 ∣ (𝐹‘𝑦) < -∞} = ∅ ↔ ∀𝑦 ∈ 𝐴 ¬ (𝐹‘𝑦) < -∞) | |
25 | 23, 24 | sylibr 237 | . 2 ⊢ (𝜑 → {𝑦 ∈ 𝐴 ∣ (𝐹‘𝑦) < -∞} = ∅) |
26 | 13, 25 | eqtrd 2774 | 1 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) < -∞} = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 399 = wceq 1542 ∈ wcel 2114 Ⅎwnfc 2880 ∀wral 3054 {crab 3058 ∅c0 4212 class class class wbr 5031 ⟶wf 6336 ‘cfv 6340 ℝcr 10617 -∞cmnf 10754 ℝ*cxr 10755 < clt 10756 ≤ cle 10757 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2711 ax-sep 5168 ax-nul 5175 ax-pow 5233 ax-pr 5297 ax-un 7482 ax-cnex 10674 ax-resscn 10675 ax-pre-lttri 10692 ax-pre-lttrn 10693 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2541 df-eu 2571 df-clab 2718 df-cleq 2731 df-clel 2812 df-nfc 2882 df-ne 2936 df-nel 3040 df-ral 3059 df-rex 3060 df-rab 3063 df-v 3401 df-sbc 3682 df-csb 3792 df-dif 3847 df-un 3849 df-in 3851 df-ss 3861 df-nul 4213 df-if 4416 df-pw 4491 df-sn 4518 df-pr 4520 df-op 4524 df-uni 4798 df-br 5032 df-opab 5094 df-mpt 5112 df-id 5430 df-po 5443 df-so 5444 df-xp 5532 df-rel 5533 df-cnv 5534 df-co 5535 df-dm 5536 df-rn 5537 df-res 5538 df-ima 5539 df-iota 6298 df-fun 6342 df-fn 6343 df-f 6344 df-f1 6345 df-fo 6346 df-f1o 6347 df-fv 6348 df-er 8323 df-en 8559 df-dom 8560 df-sdom 8561 df-pnf 10758 df-mnf 10759 df-xr 10760 df-ltxr 10761 df-le 10762 |
This theorem is referenced by: smfpimltxr 43845 |
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