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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 1912 | . . 3 ⊢ Ⅎ𝑦(𝐹‘𝑥) < -∞ | |
4 | pimltmnf2f.1 | . . . . 5 ⊢ Ⅎ𝑥𝐹 | |
5 | nfcv 2903 | . . . . 5 ⊢ Ⅎ𝑥𝑦 | |
6 | 4, 5 | nffv 6917 | . . . 4 ⊢ Ⅎ𝑥(𝐹‘𝑦) |
7 | nfcv 2903 | . . . 4 ⊢ Ⅎ𝑥 < | |
8 | nfcv 2903 | . . . 4 ⊢ Ⅎ𝑥-∞ | |
9 | 6, 7, 8 | nfbr 5195 | . . 3 ⊢ Ⅎ𝑥(𝐹‘𝑦) < -∞ |
10 | fveq2 6907 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝐹‘𝑥) = (𝐹‘𝑦)) | |
11 | 10 | breq1d 5158 | . . 3 ⊢ (𝑥 = 𝑦 → ((𝐹‘𝑥) < -∞ ↔ (𝐹‘𝑦) < -∞)) |
12 | 1, 2, 3, 9, 11 | cbvrabw 3471 | . 2 ⊢ {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) < -∞} = {𝑦 ∈ 𝐴 ∣ (𝐹‘𝑦) < -∞} |
13 | pimltmnf2f.3 | . . . . . . . 8 ⊢ (𝜑 → 𝐹:𝐴⟶ℝ) | |
14 | 13 | ffvelcdmda 7104 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → (𝐹‘𝑦) ∈ ℝ) |
15 | 14 | rexrd 11309 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → (𝐹‘𝑦) ∈ ℝ*) |
16 | 15 | mnfled 13175 | . . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → -∞ ≤ (𝐹‘𝑦)) |
17 | mnfxr 11316 | . . . . . . 7 ⊢ -∞ ∈ ℝ* | |
18 | 17 | a1i 11 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → -∞ ∈ ℝ*) |
19 | 18, 15 | xrlenltd 11325 | . . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → (-∞ ≤ (𝐹‘𝑦) ↔ ¬ (𝐹‘𝑦) < -∞)) |
20 | 16, 19 | mpbid 232 | . . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → ¬ (𝐹‘𝑦) < -∞) |
21 | 20 | ralrimiva 3144 | . . 3 ⊢ (𝜑 → ∀𝑦 ∈ 𝐴 ¬ (𝐹‘𝑦) < -∞) |
22 | rabeq0 4394 | . . 3 ⊢ ({𝑦 ∈ 𝐴 ∣ (𝐹‘𝑦) < -∞} = ∅ ↔ ∀𝑦 ∈ 𝐴 ¬ (𝐹‘𝑦) < -∞) | |
23 | 21, 22 | sylibr 234 | . 2 ⊢ (𝜑 → {𝑦 ∈ 𝐴 ∣ (𝐹‘𝑦) < -∞} = ∅) |
24 | 12, 23 | eqtrid 2787 | 1 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) < -∞} = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2106 Ⅎwnfc 2888 ∀wral 3059 {crab 3433 ∅c0 4339 class class class wbr 5148 ⟶wf 6559 ‘cfv 6563 ℝcr 11152 -∞cmnf 11291 ℝ*cxr 11292 < clt 11293 ≤ cle 11294 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 |
This theorem is referenced by: pimltmnf2 46654 smfpimltxr 46703 |
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