Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > pimgtpnf2 | Structured version Visualization version GIF version |
Description: Given a real-valued function, the preimage of an open interval, unbounded above, with lower bound +∞, is the empty set. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
Ref | Expression |
---|---|
pimgtpnf2.1 | ⊢ Ⅎ𝑥𝐹 |
pimgtpnf2.2 | ⊢ (𝜑 → 𝐹:𝐴⟶ℝ) |
Ref | Expression |
---|---|
pimgtpnf2 | ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ +∞ < (𝐹‘𝑥)} = ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfcv 2899 | . . . 4 ⊢ Ⅎ𝑥𝐴 | |
2 | nfcv 2899 | . . . 4 ⊢ Ⅎ𝑦𝐴 | |
3 | nfv 1921 | . . . 4 ⊢ Ⅎ𝑦+∞ < (𝐹‘𝑥) | |
4 | nfcv 2899 | . . . . 5 ⊢ Ⅎ𝑥+∞ | |
5 | nfcv 2899 | . . . . 5 ⊢ Ⅎ𝑥 < | |
6 | pimgtpnf2.1 | . . . . . 6 ⊢ Ⅎ𝑥𝐹 | |
7 | nfcv 2899 | . . . . . 6 ⊢ Ⅎ𝑥𝑦 | |
8 | 6, 7 | nffv 6684 | . . . . 5 ⊢ Ⅎ𝑥(𝐹‘𝑦) |
9 | 4, 5, 8 | nfbr 5077 | . . . 4 ⊢ Ⅎ𝑥+∞ < (𝐹‘𝑦) |
10 | fveq2 6674 | . . . . 5 ⊢ (𝑥 = 𝑦 → (𝐹‘𝑥) = (𝐹‘𝑦)) | |
11 | 10 | breq2d 5042 | . . . 4 ⊢ (𝑥 = 𝑦 → (+∞ < (𝐹‘𝑥) ↔ +∞ < (𝐹‘𝑦))) |
12 | 1, 2, 3, 9, 11 | cbvrabw 3391 | . . 3 ⊢ {𝑥 ∈ 𝐴 ∣ +∞ < (𝐹‘𝑥)} = {𝑦 ∈ 𝐴 ∣ +∞ < (𝐹‘𝑦)} |
13 | 12 | a1i 11 | . 2 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ +∞ < (𝐹‘𝑥)} = {𝑦 ∈ 𝐴 ∣ +∞ < (𝐹‘𝑦)}) |
14 | pimgtpnf2.2 | . . . . . . . 8 ⊢ (𝜑 → 𝐹:𝐴⟶ℝ) | |
15 | 14 | ffvelrnda 6861 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → (𝐹‘𝑦) ∈ ℝ) |
16 | 15 | rexrd 10769 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → (𝐹‘𝑦) ∈ ℝ*) |
17 | pnfxr 10773 | . . . . . . 7 ⊢ +∞ ∈ ℝ* | |
18 | 17 | a1i 11 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → +∞ ∈ ℝ*) |
19 | 15 | ltpnfd 12599 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → (𝐹‘𝑦) < +∞) |
20 | 16, 18, 19 | xrltled 12626 | . . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → (𝐹‘𝑦) ≤ +∞) |
21 | 16, 18 | xrlenltd 10785 | . . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → ((𝐹‘𝑦) ≤ +∞ ↔ ¬ +∞ < (𝐹‘𝑦))) |
22 | 20, 21 | mpbid 235 | . . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → ¬ +∞ < (𝐹‘𝑦)) |
23 | 22 | ralrimiva 3096 | . . 3 ⊢ (𝜑 → ∀𝑦 ∈ 𝐴 ¬ +∞ < (𝐹‘𝑦)) |
24 | rabeq0 4273 | . . 3 ⊢ ({𝑦 ∈ 𝐴 ∣ +∞ < (𝐹‘𝑦)} = ∅ ↔ ∀𝑦 ∈ 𝐴 ¬ +∞ < (𝐹‘𝑦)) | |
25 | 23, 24 | sylibr 237 | . 2 ⊢ (𝜑 → {𝑦 ∈ 𝐴 ∣ +∞ < (𝐹‘𝑦)} = ∅) |
26 | 13, 25 | eqtrd 2773 | 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 2879 ∀wral 3053 {crab 3057 ∅c0 4211 class class class wbr 5030 ⟶wf 6335 ‘cfv 6339 ℝcr 10614 +∞cpnf 10750 ℝ*cxr 10752 < clt 10753 ≤ cle 10754 |
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 2710 ax-sep 5167 ax-nul 5174 ax-pow 5232 ax-pr 5296 ax-un 7479 ax-cnex 10671 ax-resscn 10672 ax-pre-lttri 10689 ax-pre-lttrn 10690 |
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 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-nel 3039 df-ral 3058 df-rex 3059 df-rab 3062 df-v 3400 df-sbc 3681 df-csb 3791 df-dif 3846 df-un 3848 df-in 3850 df-ss 3860 df-nul 4212 df-if 4415 df-pw 4490 df-sn 4517 df-pr 4519 df-op 4523 df-uni 4797 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5429 df-po 5442 df-so 5443 df-xp 5531 df-rel 5532 df-cnv 5533 df-co 5534 df-dm 5535 df-rn 5536 df-res 5537 df-ima 5538 df-iota 6297 df-fun 6341 df-fn 6342 df-f 6343 df-f1 6344 df-fo 6345 df-f1o 6346 df-fv 6347 df-er 8320 df-en 8556 df-dom 8557 df-sdom 8558 df-pnf 10755 df-mnf 10756 df-xr 10757 df-ltxr 10758 df-le 10759 |
This theorem is referenced by: smfpimgtxr 43854 |
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