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 2979 | . . . 4 ⊢ Ⅎ𝑥𝐴 | |
2 | nfcv 2979 | . . . 4 ⊢ Ⅎ𝑦𝐴 | |
3 | nfv 1915 | . . . 4 ⊢ Ⅎ𝑦(𝐹‘𝑥) < -∞ | |
4 | pimltmnf2.1 | . . . . . 6 ⊢ Ⅎ𝑥𝐹 | |
5 | nfcv 2979 | . . . . . 6 ⊢ Ⅎ𝑥𝑦 | |
6 | 4, 5 | nffv 6682 | . . . . 5 ⊢ Ⅎ𝑥(𝐹‘𝑦) |
7 | nfcv 2979 | . . . . 5 ⊢ Ⅎ𝑥 < | |
8 | nfcv 2979 | . . . . 5 ⊢ Ⅎ𝑥-∞ | |
9 | 6, 7, 8 | nfbr 5115 | . . . 4 ⊢ Ⅎ𝑥(𝐹‘𝑦) < -∞ |
10 | fveq2 6672 | . . . . 5 ⊢ (𝑥 = 𝑦 → (𝐹‘𝑥) = (𝐹‘𝑦)) | |
11 | 10 | breq1d 5078 | . . . 4 ⊢ (𝑥 = 𝑦 → ((𝐹‘𝑥) < -∞ ↔ (𝐹‘𝑦) < -∞)) |
12 | 1, 2, 3, 9, 11 | cbvrabw 3491 | . . 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 10700 | . . . . . . 7 ⊢ -∞ ∈ ℝ* | |
15 | 14 | a1i 11 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → -∞ ∈ ℝ*) |
16 | pimltmnf2.2 | . . . . . . . 8 ⊢ (𝜑 → 𝐹:𝐴⟶ℝ) | |
17 | 16 | ffvelrnda 6853 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → (𝐹‘𝑦) ∈ ℝ) |
18 | 17 | rexrd 10693 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → (𝐹‘𝑦) ∈ ℝ*) |
19 | 17 | mnfltd 12522 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → -∞ < (𝐹‘𝑦)) |
20 | 15, 18, 19 | xrltled 12546 | . . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → -∞ ≤ (𝐹‘𝑦)) |
21 | 15, 18 | xrlenltd 10709 | . . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → (-∞ ≤ (𝐹‘𝑦) ↔ ¬ (𝐹‘𝑦) < -∞)) |
22 | 20, 21 | mpbid 234 | . . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → ¬ (𝐹‘𝑦) < -∞) |
23 | 22 | ralrimiva 3184 | . . 3 ⊢ (𝜑 → ∀𝑦 ∈ 𝐴 ¬ (𝐹‘𝑦) < -∞) |
24 | rabeq0 4340 | . . 3 ⊢ ({𝑦 ∈ 𝐴 ∣ (𝐹‘𝑦) < -∞} = ∅ ↔ ∀𝑦 ∈ 𝐴 ¬ (𝐹‘𝑦) < -∞) | |
25 | 23, 24 | sylibr 236 | . 2 ⊢ (𝜑 → {𝑦 ∈ 𝐴 ∣ (𝐹‘𝑦) < -∞} = ∅) |
26 | 13, 25 | eqtrd 2858 | 1 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) < -∞} = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 Ⅎwnfc 2963 ∀wral 3140 {crab 3144 ∅c0 4293 class class class wbr 5068 ⟶wf 6353 ‘cfv 6357 ℝcr 10538 -∞cmnf 10675 ℝ*cxr 10676 < clt 10677 ≤ cle 10678 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-pre-lttri 10613 ax-pre-lttrn 10614 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-po 5476 df-so 5477 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 |
This theorem is referenced by: smfpimltxr 43031 |
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