| 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 2905 | . . 3 ⊢ Ⅎ𝑦𝐴 | |
| 3 | nfv 1914 | . . 3 ⊢ Ⅎ𝑦(𝐹‘𝑥) < -∞ | |
| 4 | pimltmnf2f.1 | . . . . 5 ⊢ Ⅎ𝑥𝐹 | |
| 5 | nfcv 2905 | . . . . 5 ⊢ Ⅎ𝑥𝑦 | |
| 6 | 4, 5 | nffv 6916 | . . . 4 ⊢ Ⅎ𝑥(𝐹‘𝑦) |
| 7 | nfcv 2905 | . . . 4 ⊢ Ⅎ𝑥 < | |
| 8 | nfcv 2905 | . . . 4 ⊢ Ⅎ𝑥-∞ | |
| 9 | 6, 7, 8 | nfbr 5190 | . . 3 ⊢ Ⅎ𝑥(𝐹‘𝑦) < -∞ |
| 10 | fveq2 6906 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝐹‘𝑥) = (𝐹‘𝑦)) | |
| 11 | 10 | breq1d 5153 | . . 3 ⊢ (𝑥 = 𝑦 → ((𝐹‘𝑥) < -∞ ↔ (𝐹‘𝑦) < -∞)) |
| 12 | 1, 2, 3, 9, 11 | cbvrabw 3473 | . 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 11311 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → (𝐹‘𝑦) ∈ ℝ*) |
| 16 | 15 | mnfled 13178 | . . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → -∞ ≤ (𝐹‘𝑦)) |
| 17 | mnfxr 11318 | . . . . . . 7 ⊢ -∞ ∈ ℝ* | |
| 18 | 17 | a1i 11 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → -∞ ∈ ℝ*) |
| 19 | 18, 15 | xrlenltd 11327 | . . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → (-∞ ≤ (𝐹‘𝑦) ↔ ¬ (𝐹‘𝑦) < -∞)) |
| 20 | 16, 19 | mpbid 232 | . . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → ¬ (𝐹‘𝑦) < -∞) |
| 21 | 20 | ralrimiva 3146 | . . 3 ⊢ (𝜑 → ∀𝑦 ∈ 𝐴 ¬ (𝐹‘𝑦) < -∞) |
| 22 | rabeq0 4388 | . . 3 ⊢ ({𝑦 ∈ 𝐴 ∣ (𝐹‘𝑦) < -∞} = ∅ ↔ ∀𝑦 ∈ 𝐴 ¬ (𝐹‘𝑦) < -∞) | |
| 23 | 21, 22 | sylibr 234 | . 2 ⊢ (𝜑 → {𝑦 ∈ 𝐴 ∣ (𝐹‘𝑦) < -∞} = ∅) |
| 24 | 12, 23 | eqtrid 2789 | 1 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) < -∞} = ∅) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 Ⅎwnfc 2890 ∀wral 3061 {crab 3436 ∅c0 4333 class class class wbr 5143 ⟶wf 6557 ‘cfv 6561 ℝcr 11154 -∞cmnf 11293 ℝ*cxr 11294 < clt 11295 ≤ cle 11296 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 |
| This theorem is referenced by: pimltmnf2 46713 smfpimltxr 46762 |
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