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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 44381 | . . 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 13109 | . . . . . . . 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 3446 | . . . . . 6 ⊢ (𝑥 ∈ {𝑥 ∈ 𝐴 ∣ -∞ < 𝐵} ↔ (𝑥 ∈ 𝐴 ∧ -∞ < 𝐵)) | |
11 | 9, 10 | sylibr 233 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ -∞ < 𝐵}) |
12 | 11 | ex 412 | . . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ -∞ < 𝐵})) |
13 | 4, 12 | ralrimi 3248 | . . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ -∞ < 𝐵}) |
14 | nfrab1 3445 | . . . 4 ⊢ Ⅎ𝑥{𝑥 ∈ 𝐴 ∣ -∞ < 𝐵} | |
15 | 1, 14 | dfss3f 3968 | . . 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 3994 | 1 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ -∞ < 𝐵} = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1533 Ⅎwnf 1777 ∈ wcel 2098 Ⅎwnfc 2877 ∀wral 3055 {crab 3426 ⊆ wss 3943 class class class wbr 5141 ℝcr 11111 -∞cmnf 11250 < clt 11252 |
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 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 ax-cnex 11168 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ral 3056 df-rex 3065 df-rab 3427 df-v 3470 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-br 5142 df-opab 5204 df-xp 5675 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 |
This theorem is referenced by: pimgtmnf 46011 |
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