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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > pimgtmnff | 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 whole domain. (Contributed by Glauco Siliprandi, 20-Dec-2024.) |
Ref | Expression |
---|---|
pimgtmnff.1 | ⊢ Ⅎ𝑥𝜑 |
pimgtmnff.2 | ⊢ Ⅎ𝑥𝐴 |
pimgtmnff.3 | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ) |
Ref | Expression |
---|---|
pimgtmnff | ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ -∞ < 𝐵} = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pimgtmnff.2 | . . . 4 ⊢ Ⅎ𝑥𝐴 | |
2 | 1 | ssrab2f 44548 | . . 3 ⊢ {𝑥 ∈ 𝐴 ∣ -∞ < 𝐵} ⊆ 𝐴 |
3 | 2 | a1i 11 | . 2 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ -∞ < 𝐵} ⊆ 𝐴) |
4 | pimgtmnff.1 | . . . 4 ⊢ Ⅎ𝑥𝜑 | |
5 | simpr 483 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐴) | |
6 | pimgtmnff.3 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ) | |
7 | mnflt 13135 | . . . . . . . 8 ⊢ (𝐵 ∈ ℝ → -∞ < 𝐵) | |
8 | 6, 7 | syl 17 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → -∞ < 𝐵) |
9 | 5, 8 | jca 510 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑥 ∈ 𝐴 ∧ -∞ < 𝐵)) |
10 | rabid 3440 | . . . . . 6 ⊢ (𝑥 ∈ {𝑥 ∈ 𝐴 ∣ -∞ < 𝐵} ↔ (𝑥 ∈ 𝐴 ∧ -∞ < 𝐵)) | |
11 | 9, 10 | sylibr 233 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ -∞ < 𝐵}) |
12 | 11 | ex 411 | . . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ -∞ < 𝐵})) |
13 | 4, 12 | ralrimi 3245 | . . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ -∞ < 𝐵}) |
14 | nfrab1 3439 | . . . 4 ⊢ Ⅎ𝑥{𝑥 ∈ 𝐴 ∣ -∞ < 𝐵} | |
15 | 1, 14 | dfss3f 3963 | . . 3 ⊢ (𝐴 ⊆ {𝑥 ∈ 𝐴 ∣ -∞ < 𝐵} ↔ ∀𝑥 ∈ 𝐴 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ -∞ < 𝐵}) |
16 | 13, 15 | sylibr 233 | . 2 ⊢ (𝜑 → 𝐴 ⊆ {𝑥 ∈ 𝐴 ∣ -∞ < 𝐵}) |
17 | 3, 16 | eqssd 3990 | 1 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ -∞ < 𝐵} = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1533 Ⅎwnf 1777 ∈ wcel 2098 Ⅎwnfc 2875 ∀wral 3051 {crab 3419 ⊆ wss 3939 class class class wbr 5143 ℝcr 11137 -∞cmnf 11276 < clt 11278 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7738 ax-cnex 11194 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3465 df-dif 3942 df-un 3944 df-ss 3956 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-br 5144 df-opab 5206 df-xp 5678 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 |
This theorem is referenced by: pimgtmnf 46174 |
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