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 2751 | . . . 4 ⊢ Ⅎ𝑥𝐴 | |
2 | nfcv 2751 | . . . 4 ⊢ Ⅎ𝑦𝐴 | |
3 | nfv 1830 | . . . 4 ⊢ Ⅎ𝑦+∞ < (𝐹‘𝑥) | |
4 | nfcv 2751 | . . . . 5 ⊢ Ⅎ𝑥+∞ | |
5 | nfcv 2751 | . . . . 5 ⊢ Ⅎ𝑥 < | |
6 | pimgtpnf2.1 | . . . . . 6 ⊢ Ⅎ𝑥𝐹 | |
7 | nfcv 2751 | . . . . . 6 ⊢ Ⅎ𝑥𝑦 | |
8 | 6, 7 | nffv 6110 | . . . . 5 ⊢ Ⅎ𝑥(𝐹‘𝑦) |
9 | 4, 5, 8 | nfbr 4629 | . . . 4 ⊢ Ⅎ𝑥+∞ < (𝐹‘𝑦) |
10 | fveq2 6103 | . . . . 5 ⊢ (𝑥 = 𝑦 → (𝐹‘𝑥) = (𝐹‘𝑦)) | |
11 | 10 | breq2d 4595 | . . . 4 ⊢ (𝑥 = 𝑦 → (+∞ < (𝐹‘𝑥) ↔ +∞ < (𝐹‘𝑦))) |
12 | 1, 2, 3, 9, 11 | cbvrab 3171 | . . 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 6267 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → (𝐹‘𝑦) ∈ ℝ) |
16 | 15 | rexrd 9968 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → (𝐹‘𝑦) ∈ ℝ*) |
17 | pnfxr 9971 | . . . . . . 7 ⊢ +∞ ∈ ℝ* | |
18 | 17 | a1i 11 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → +∞ ∈ ℝ*) |
19 | 15 | ltpnfd 11831 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → (𝐹‘𝑦) < +∞) |
20 | 16, 18, 19 | xrltled 38427 | . . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → (𝐹‘𝑦) ≤ +∞) |
21 | 16, 18 | xrlenltd 9983 | . . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → ((𝐹‘𝑦) ≤ +∞ ↔ ¬ +∞ < (𝐹‘𝑦))) |
22 | 20, 21 | mpbid 221 | . . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → ¬ +∞ < (𝐹‘𝑦)) |
23 | 22 | ralrimiva 2949 | . . 3 ⊢ (𝜑 → ∀𝑦 ∈ 𝐴 ¬ +∞ < (𝐹‘𝑦)) |
24 | rabeq0 3911 | . . 3 ⊢ ({𝑦 ∈ 𝐴 ∣ +∞ < (𝐹‘𝑦)} = ∅ ↔ ∀𝑦 ∈ 𝐴 ¬ +∞ < (𝐹‘𝑦)) | |
25 | 23, 24 | sylibr 223 | . 2 ⊢ (𝜑 → {𝑦 ∈ 𝐴 ∣ +∞ < (𝐹‘𝑦)} = ∅) |
26 | 13, 25 | eqtrd 2644 | 1 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ +∞ < (𝐹‘𝑥)} = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 Ⅎwnfc 2738 ∀wral 2896 {crab 2900 ∅c0 3874 class class class wbr 4583 ⟶wf 5800 ‘cfv 5804 ℝcr 9814 +∞cpnf 9950 ℝ*cxr 9952 < clt 9953 ≤ cle 9954 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-pre-lttri 9889 ax-pre-lttrn 9890 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-po 4959 df-so 4960 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 |
This theorem is referenced by: smfpimgtxr 39666 |
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