![]() |
Mathbox for Glauco Siliprandi |
< Previous
Next >
Nearby theorems |
|
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 481 | . . . . . 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 481 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ) |
8 | 5, 7 | fvmpt2d 7011 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = 𝐵) |
9 | 3, 8 | breqtrrd 5176 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ≤ (𝐹‘𝑥)) |
10 | pimconstlt0.c | . . . . . . 7 ⊢ (𝜑 → 𝐶 ∈ ℝ*) | |
11 | 10 | adantr 481 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ ℝ*) |
12 | 8, 7 | eqeltrd 2833 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) ∈ ℝ) |
13 | 12 | rexrd 11263 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) ∈ ℝ*) |
14 | 11, 13 | xrlenltd 11279 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐶 ≤ (𝐹‘𝑥) ↔ ¬ (𝐹‘𝑥) < 𝐶)) |
15 | 9, 14 | mpbid 231 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ¬ (𝐹‘𝑥) < 𝐶) |
16 | 15 | ex 413 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → ¬ (𝐹‘𝑥) < 𝐶)) |
17 | 1, 16 | ralrimi 3254 | . 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ¬ (𝐹‘𝑥) < 𝐶) |
18 | rabeq0 4384 | . 2 ⊢ ({𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) < 𝐶} = ∅ ↔ ∀𝑥 ∈ 𝐴 ¬ (𝐹‘𝑥) < 𝐶) | |
19 | 17, 18 | sylibr 233 | 1 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) < 𝐶} = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 = wceq 1541 Ⅎwnf 1785 ∈ wcel 2106 ∀wral 3061 {crab 3432 ∅c0 4322 class class class wbr 5148 ↦ cmpt 5231 ‘cfv 6543 ℝcr 11108 ℝ*cxr 11246 < clt 11247 ≤ cle 11248 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5299 ax-nul 5306 ax-pr 5427 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-iota 6495 df-fun 6545 df-fv 6551 df-xr 11251 df-le 11253 |
This theorem is referenced by: smfconst 45455 |
Copyright terms: Public domain | W3C validator |