|   | Mathbox for Glauco Siliprandi | < Previous  
      Next > Nearby theorems | |
| Mirrors > Home > MPE Home > Th. List > Mathboxes > pimltpnff | 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, 20-Dec-2024.) | 
| Ref | Expression | 
|---|---|
| pimltpnff.1 | ⊢ Ⅎ𝑥𝜑 | 
| pimltpnff.2 | ⊢ Ⅎ𝑥𝐴 | 
| pimltpnff.3 | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ) | 
| Ref | Expression | 
|---|---|
| pimltpnff | ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝐵 < +∞} = 𝐴) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | pimltpnff.2 | . . . 4 ⊢ Ⅎ𝑥𝐴 | |
| 2 | 1 | ssrab2f 45127 | . . 3 ⊢ {𝑥 ∈ 𝐴 ∣ 𝐵 < +∞} ⊆ 𝐴 | 
| 3 | 2 | a1i 11 | . 2 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝐵 < +∞} ⊆ 𝐴) | 
| 4 | pimltpnff.1 | . . . 4 ⊢ Ⅎ𝑥𝜑 | |
| 5 | simpr 484 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐴) | |
| 6 | pimltpnff.3 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ) | |
| 7 | ltpnf 13163 | . . . . . . . 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 3457 | . . . . . 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 3256 | . . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝐵 < +∞}) | 
| 14 | nfrab1 3456 | . . . 4 ⊢ Ⅎ𝑥{𝑥 ∈ 𝐴 ∣ 𝐵 < +∞} | |
| 15 | 1, 14 | dfss3f 3974 | . . 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 4000 | 1 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝐵 < +∞} = 𝐴) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1539 Ⅎwnf 1782 ∈ wcel 2107 Ⅎwnfc 2889 ∀wral 3060 {crab 3435 ⊆ wss 3950 class class class wbr 5142 ℝcr 11155 +∞cpnf 11293 < clt 11296 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 ax-cnex 11212 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-ss 3967 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-br 5143 df-opab 5205 df-xp 5690 df-pnf 11298 df-xr 11300 df-ltxr 11301 | 
| This theorem is referenced by: pimltpnf 46724 | 
| Copyright terms: Public domain | W3C validator |