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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > pimltpnf | Structured version Visualization version GIF version |
Description: Given a real-valued function, the preimage of an open interval, unbounded below, with upper bound +∞, is the whole domain. (Contributed by Glauco Siliprandi, 26-Jun-2021.) (Revised by Glauco Siliprandi, 20-Dec-2024.) |
Ref | Expression |
---|---|
pimltpnf.1 | ⊢ Ⅎ𝑥𝜑 |
pimltpnf.2 | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ) |
Ref | Expression |
---|---|
pimltpnf | ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝐵 < +∞} = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pimltpnf.1 | . 2 ⊢ Ⅎ𝑥𝜑 | |
2 | nfcv 2897 | . 2 ⊢ Ⅎ𝑥𝐴 | |
3 | pimltpnf.2 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ) | |
4 | 1, 2, 3 | pimltpnff 45973 | 1 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝐵 < +∞} = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1533 Ⅎwnf 1777 ∈ wcel 2098 {crab 3426 class class class wbr 5141 ℝcr 11108 +∞cpnf 11246 < clt 11249 |
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 7721 ax-cnex 11165 |
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 11251 df-xr 11253 df-ltxr 11254 |
This theorem is referenced by: pimltpnf2f 45982 |
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