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 483 | . . . . . 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 483 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ) |
8 | 5, 7 | fvmpt2d 6784 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = 𝐵) |
9 | 3, 8 | breqtrrd 5097 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ≤ (𝐹‘𝑥)) |
10 | pimconstlt0.c | . . . . . . 7 ⊢ (𝜑 → 𝐶 ∈ ℝ*) | |
11 | 10 | adantr 483 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ ℝ*) |
12 | 8, 7 | eqeltrd 2916 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) ∈ ℝ) |
13 | 12 | rexrd 10694 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) ∈ ℝ*) |
14 | 11, 13 | xrlenltd 10710 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐶 ≤ (𝐹‘𝑥) ↔ ¬ (𝐹‘𝑥) < 𝐶)) |
15 | 9, 14 | mpbid 234 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ¬ (𝐹‘𝑥) < 𝐶) |
16 | 15 | ex 415 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → ¬ (𝐹‘𝑥) < 𝐶)) |
17 | 1, 16 | ralrimi 3219 | . 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ¬ (𝐹‘𝑥) < 𝐶) |
18 | rabeq0 4341 | . 2 ⊢ ({𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) < 𝐶} = ∅ ↔ ∀𝑥 ∈ 𝐴 ¬ (𝐹‘𝑥) < 𝐶) | |
19 | 17, 18 | sylibr 236 | 1 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) < 𝐶} = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 398 = wceq 1536 Ⅎwnf 1783 ∈ wcel 2113 ∀wral 3141 {crab 3145 ∅c0 4294 class class class wbr 5069 ↦ cmpt 5149 ‘cfv 6358 ℝcr 10539 ℝ*cxr 10677 < clt 10678 ≤ cle 10679 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ral 3146 df-rex 3147 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-sn 4571 df-pr 4573 df-op 4577 df-uni 4842 df-br 5070 df-opab 5132 df-mpt 5150 df-id 5463 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-iota 6317 df-fun 6360 df-fv 6366 df-xr 10682 df-le 10684 |
This theorem is referenced by: smfconst 43033 |
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