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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > pimconstlt0 | Structured version Visualization version GIF version |
Description: Given a constant function, its preimage with respect to an unbounded below, open interval, with upper bound less than or equal to the constant, is the empty set. Second part of Proposition 121E (a) of [Fremlin1] p. 37 . (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
Ref | Expression |
---|---|
pimconstlt0.x | ⊢ Ⅎ𝑥𝜑 |
pimconstlt0.b | ⊢ (𝜑 → 𝐵 ∈ ℝ) |
pimconstlt0.f | ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) |
pimconstlt0.c | ⊢ (𝜑 → 𝐶 ∈ ℝ*) |
pimconstlt0.l | ⊢ (𝜑 → 𝐶 ≤ 𝐵) |
Ref | Expression |
---|---|
pimconstlt0 | ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) < 𝐶} = ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pimconstlt0.x | . . 3 ⊢ Ⅎ𝑥𝜑 | |
2 | pimconstlt0.l | . . . . . . 7 ⊢ (𝜑 → 𝐶 ≤ 𝐵) | |
3 | 2 | adantr 474 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ≤ 𝐵) |
4 | pimconstlt0.f | . . . . . . . 8 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) | |
5 | 4 | a1i 11 | . . . . . . 7 ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)) |
6 | pimconstlt0.b | . . . . . . . 8 ⊢ (𝜑 → 𝐵 ∈ ℝ) | |
7 | 6 | adantr 474 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ) |
8 | 5, 7 | fvmpt2d 6540 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = 𝐵) |
9 | 3, 8 | breqtrrd 4901 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ≤ (𝐹‘𝑥)) |
10 | pimconstlt0.c | . . . . . . 7 ⊢ (𝜑 → 𝐶 ∈ ℝ*) | |
11 | 10 | adantr 474 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ ℝ*) |
12 | 8, 7 | eqeltrd 2906 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) ∈ ℝ) |
13 | 12 | rexrd 10406 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) ∈ ℝ*) |
14 | 11, 13 | xrlenltd 10423 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐶 ≤ (𝐹‘𝑥) ↔ ¬ (𝐹‘𝑥) < 𝐶)) |
15 | 9, 14 | mpbid 224 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ¬ (𝐹‘𝑥) < 𝐶) |
16 | 15 | ex 403 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → ¬ (𝐹‘𝑥) < 𝐶)) |
17 | 1, 16 | ralrimi 3166 | . 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ¬ (𝐹‘𝑥) < 𝐶) |
18 | rabeq0 4186 | . 2 ⊢ ({𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) < 𝐶} = ∅ ↔ ∀𝑥 ∈ 𝐴 ¬ (𝐹‘𝑥) < 𝐶) | |
19 | 17, 18 | sylibr 226 | 1 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) < 𝐶} = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 386 = wceq 1658 Ⅎwnf 1884 ∈ wcel 2166 ∀wral 3117 {crab 3121 ∅c0 4144 class class class wbr 4873 ↦ cmpt 4952 ‘cfv 6123 ℝcr 10251 ℝ*cxr 10390 < clt 10391 ≤ cle 10392 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2803 ax-sep 5005 ax-nul 5013 ax-pow 5065 ax-pr 5127 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ral 3122 df-rex 3123 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4145 df-if 4307 df-sn 4398 df-pr 4400 df-op 4404 df-uni 4659 df-br 4874 df-opab 4936 df-mpt 4953 df-id 5250 df-xp 5348 df-rel 5349 df-cnv 5350 df-co 5351 df-dm 5352 df-rn 5353 df-res 5354 df-ima 5355 df-iota 6086 df-fun 6125 df-fv 6131 df-xr 10395 df-le 10397 |
This theorem is referenced by: smfconst 41752 |
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