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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 4079 | . . 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 484 | . . . . . . 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 480 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ) | 
| 9 | 6, 8 | fvmpt2d 7028 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = 𝐵) | 
| 10 | pimconstlt1.4 | . . . . . . . . 9 ⊢ (𝜑 → 𝐵 < 𝐶) | |
| 11 | 10 | adantr 480 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 < 𝐶) | 
| 12 | 9, 11 | eqbrtrd 5164 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) < 𝐶) | 
| 13 | 4, 12 | jca 511 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑥 ∈ 𝐴 ∧ (𝐹‘𝑥) < 𝐶)) | 
| 14 | rabid 3457 | . . . . . 6 ⊢ (𝑥 ∈ {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) < 𝐶} ↔ (𝑥 ∈ 𝐴 ∧ (𝐹‘𝑥) < 𝐶)) | |
| 15 | 13, 14 | sylibr 234 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) < 𝐶}) | 
| 16 | 15 | ex 412 | . . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) < 𝐶})) | 
| 17 | 3, 16 | ralrimi 3256 | . . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) < 𝐶}) | 
| 18 | nfcv 2904 | . . . 4 ⊢ Ⅎ𝑥𝐴 | |
| 19 | nfrab1 3456 | . . . 4 ⊢ Ⅎ𝑥{𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) < 𝐶} | |
| 20 | 18, 19 | dfss3f 3974 | . . 3 ⊢ (𝐴 ⊆ {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) < 𝐶} ↔ ∀𝑥 ∈ 𝐴 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) < 𝐶}) | 
| 21 | 17, 20 | sylibr 234 | . 2 ⊢ (𝜑 → 𝐴 ⊆ {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) < 𝐶}) | 
| 22 | 2, 21 | eqssd 4000 | 1 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) < 𝐶} = 𝐴) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1539 Ⅎwnf 1782 ∈ wcel 2107 ∀wral 3060 {crab 3435 ⊆ wss 3950 class class class wbr 5142 ↦ cmpt 5224 ‘cfv 6560 ℝcr 11155 < clt 11296 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pr 5431 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-br 5143 df-opab 5205 df-mpt 5225 df-id 5577 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-iota 6513 df-fun 6562 df-fv 6568 | 
| This theorem is referenced by: smfconst 46769 | 
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