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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 2905 | . 2 ⊢ Ⅎ𝑥𝐴 | |
| 3 | pimltpnf.2 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ) | |
| 4 | 1, 2, 3 | pimltpnff 46718 | 1 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝐵 < +∞} = 𝐴) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 Ⅎwnf 1783 ∈ wcel 2108 {crab 3436 class class class wbr 5143 ℝcr 11154 +∞cpnf 11292 < clt 11295 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-xp 5691 df-pnf 11297 df-xr 11299 df-ltxr 11300 | 
| This theorem is referenced by: pimltpnf2f 46727 | 
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