Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > pimconstlt1 | Structured version Visualization version GIF version |
Description: Given a constant function, its preimage with respect to an unbounded below, open interval, with upper bound larger than the constant, is the whole domain. First part of Proposition 121E (a) of [Fremlin1] p. 37 . (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
Ref | Expression |
---|---|
pimconstlt1.1 | ⊢ Ⅎ𝑥𝜑 |
pimconstlt1.2 | ⊢ (𝜑 → 𝐵 ∈ ℝ) |
pimconstlt1.3 | ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) |
pimconstlt1.4 | ⊢ (𝜑 → 𝐵 < 𝐶) |
Ref | Expression |
---|---|
pimconstlt1 | ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) < 𝐶} = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssrab2 4058 | . . 3 ⊢ {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) < 𝐶} ⊆ 𝐴 | |
2 | 1 | a1i 11 | . 2 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) < 𝐶} ⊆ 𝐴) |
3 | pimconstlt1.1 | . . . 4 ⊢ Ⅎ𝑥𝜑 | |
4 | simpr 487 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐴) | |
5 | pimconstlt1.3 | . . . . . . . . . 10 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) | |
6 | 5 | a1i 11 | . . . . . . . . 9 ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)) |
7 | pimconstlt1.2 | . . . . . . . . . 10 ⊢ (𝜑 → 𝐵 ∈ ℝ) | |
8 | 7 | adantr 483 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ) |
9 | 6, 8 | fvmpt2d 6783 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = 𝐵) |
10 | pimconstlt1.4 | . . . . . . . . 9 ⊢ (𝜑 → 𝐵 < 𝐶) | |
11 | 10 | adantr 483 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 < 𝐶) |
12 | 9, 11 | eqbrtrd 5090 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) < 𝐶) |
13 | 4, 12 | jca 514 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑥 ∈ 𝐴 ∧ (𝐹‘𝑥) < 𝐶)) |
14 | rabid 3380 | . . . . . 6 ⊢ (𝑥 ∈ {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) < 𝐶} ↔ (𝑥 ∈ 𝐴 ∧ (𝐹‘𝑥) < 𝐶)) | |
15 | 13, 14 | sylibr 236 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) < 𝐶}) |
16 | 15 | ex 415 | . . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) < 𝐶})) |
17 | 3, 16 | ralrimi 3218 | . . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) < 𝐶}) |
18 | nfcv 2979 | . . . 4 ⊢ Ⅎ𝑥𝐴 | |
19 | nfrab1 3386 | . . . 4 ⊢ Ⅎ𝑥{𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) < 𝐶} | |
20 | 18, 19 | dfss3f 3961 | . . 3 ⊢ (𝐴 ⊆ {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) < 𝐶} ↔ ∀𝑥 ∈ 𝐴 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) < 𝐶}) |
21 | 17, 20 | sylibr 236 | . 2 ⊢ (𝜑 → 𝐴 ⊆ {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) < 𝐶}) |
22 | 2, 21 | eqssd 3986 | 1 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) < 𝐶} = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 Ⅎwnf 1784 ∈ wcel 2114 ∀wral 3140 {crab 3144 ⊆ wss 3938 class class class wbr 5068 ↦ cmpt 5148 ‘cfv 6357 ℝcr 10538 < clt 10677 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-iota 6316 df-fun 6359 df-fv 6365 |
This theorem is referenced by: smfconst 43033 |
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