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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > pimgtmnf | 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, 26-Jun-2021.) |
Ref | Expression |
---|---|
pimgtmnf.1 | ⊢ Ⅎ𝑥𝜑 |
pimgtmnf.2 | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ) |
Ref | Expression |
---|---|
pimgtmnf | ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ -∞ < 𝐵} = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pimgtmnf.1 | . . 3 ⊢ Ⅎ𝑥𝜑 | |
2 | eqidd 2772 | . . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵)) | |
3 | pimgtmnf.2 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ) | |
4 | 2, 3 | fvmpt2d 6437 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) = 𝐵) |
5 | 4 | eqcomd 2777 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 = ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥)) |
6 | 5 | breq2d 4799 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (-∞ < 𝐵 ↔ -∞ < ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥))) |
7 | 1, 6 | rabbida 39793 | . 2 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ -∞ < 𝐵} = {𝑥 ∈ 𝐴 ∣ -∞ < ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥)}) |
8 | nfmpt1 4882 | . . 3 ⊢ Ⅎ𝑥(𝑥 ∈ 𝐴 ↦ 𝐵) | |
9 | eqid 2771 | . . . 4 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵) | |
10 | 1, 3, 9 | fmptdf 6532 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵):𝐴⟶ℝ) |
11 | 8, 10 | pimgtmnf2 41439 | . 2 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ -∞ < ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥)} = 𝐴) |
12 | 7, 11 | eqtrd 2805 | 1 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ -∞ < 𝐵} = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 382 = wceq 1631 Ⅎwnf 1856 ∈ wcel 2145 {crab 3065 class class class wbr 4787 ↦ cmpt 4864 ‘cfv 6030 ℝcr 10141 -∞cmnf 10278 < clt 10280 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-sep 4916 ax-nul 4924 ax-pow 4975 ax-pr 5035 ax-un 7100 ax-cnex 10198 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 837 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-ral 3066 df-rex 3067 df-rab 3070 df-v 3353 df-sbc 3588 df-csb 3683 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-nul 4064 df-if 4227 df-pw 4300 df-sn 4318 df-pr 4320 df-op 4324 df-uni 4576 df-br 4788 df-opab 4848 df-mpt 4865 df-id 5158 df-xp 5256 df-rel 5257 df-cnv 5258 df-co 5259 df-dm 5260 df-rn 5261 df-res 5262 df-ima 5263 df-iota 5993 df-fun 6032 df-fn 6033 df-f 6034 df-fv 6038 df-pnf 10282 df-mnf 10283 df-xr 10284 df-ltxr 10285 |
This theorem is referenced by: smfpimgtxr 41503 |
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