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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 44954 | . . 3 ⊢ {𝑥 ∈ 𝐴 ∣ -∞ < 𝐵} ⊆ 𝐴 |
3 | 2 | a1i 11 | . 2 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ -∞ < 𝐵} ⊆ 𝐴) |
4 | pimgtmnff.1 | . . . 4 ⊢ Ⅎ𝑥𝜑 | |
5 | simpr 484 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐴) | |
6 | pimgtmnff.3 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ) | |
7 | mnflt 13182 | . . . . . . . 8 ⊢ (𝐵 ∈ ℝ → -∞ < 𝐵) | |
8 | 6, 7 | syl 17 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → -∞ < 𝐵) |
9 | 5, 8 | jca 511 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑥 ∈ 𝐴 ∧ -∞ < 𝐵)) |
10 | rabid 3459 | . . . . . 6 ⊢ (𝑥 ∈ {𝑥 ∈ 𝐴 ∣ -∞ < 𝐵} ↔ (𝑥 ∈ 𝐴 ∧ -∞ < 𝐵)) | |
11 | 9, 10 | sylibr 234 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ -∞ < 𝐵}) |
12 | 11 | ex 412 | . . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ -∞ < 𝐵})) |
13 | 4, 12 | ralrimi 3258 | . . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ -∞ < 𝐵}) |
14 | nfrab1 3458 | . . . 4 ⊢ Ⅎ𝑥{𝑥 ∈ 𝐴 ∣ -∞ < 𝐵} | |
15 | 1, 14 | dfss3f 3994 | . . 3 ⊢ (𝐴 ⊆ {𝑥 ∈ 𝐴 ∣ -∞ < 𝐵} ↔ ∀𝑥 ∈ 𝐴 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ -∞ < 𝐵}) |
16 | 13, 15 | sylibr 234 | . 2 ⊢ (𝜑 → 𝐴 ⊆ {𝑥 ∈ 𝐴 ∣ -∞ < 𝐵}) |
17 | 3, 16 | eqssd 4020 | 1 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ -∞ < 𝐵} = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 Ⅎwnf 1781 ∈ wcel 2103 Ⅎwnfc 2888 ∀wral 3063 {crab 3438 ⊆ wss 3970 class class class wbr 5169 ℝcr 11179 -∞cmnf 11318 < clt 11320 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2105 ax-9 2113 ax-10 2136 ax-11 2153 ax-12 2173 ax-ext 2705 ax-sep 5320 ax-nul 5327 ax-pow 5386 ax-pr 5450 ax-un 7766 ax-cnex 11236 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2890 df-ral 3064 df-rex 3073 df-rab 3439 df-v 3484 df-dif 3973 df-un 3975 df-ss 3987 df-nul 4348 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-br 5170 df-opab 5232 df-xp 5705 df-pnf 11322 df-mnf 11323 df-xr 11324 df-ltxr 11325 |
This theorem is referenced by: pimgtmnf 46579 |
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