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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > rnmptbddlem | Structured version Visualization version GIF version |
Description: Boundness of the range of a function in maps-to notation. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
Ref | Expression |
---|---|
rnmptbddlem.x | ⊢ Ⅎ𝑥𝜑 |
rnmptbddlem.b | ⊢ (𝜑 → ∃𝑦 ∈ ℝ ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑦) |
Ref | Expression |
---|---|
rnmptbddlem | ⊢ (𝜑 → ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑧 ≤ 𝑦) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2725 | . . . . . 6 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵) | |
2 | 1 | elrnmpt 5958 | . . . . 5 ⊢ (𝑧 ∈ V → (𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵) ↔ ∃𝑥 ∈ 𝐴 𝑧 = 𝐵)) |
3 | 2 | elv 3467 | . . . 4 ⊢ (𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵) ↔ ∃𝑥 ∈ 𝐴 𝑧 = 𝐵) |
4 | rnmptbddlem.x | . . . . . . . 8 ⊢ Ⅎ𝑥𝜑 | |
5 | nfv 1909 | . . . . . . . 8 ⊢ Ⅎ𝑥 𝑦 ∈ ℝ | |
6 | 4, 5 | nfan 1894 | . . . . . . 7 ⊢ Ⅎ𝑥(𝜑 ∧ 𝑦 ∈ ℝ) |
7 | nfra1 3271 | . . . . . . 7 ⊢ Ⅎ𝑥∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑦 | |
8 | 6, 7 | nfan 1894 | . . . . . 6 ⊢ Ⅎ𝑥((𝜑 ∧ 𝑦 ∈ ℝ) ∧ ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑦) |
9 | nfv 1909 | . . . . . 6 ⊢ Ⅎ𝑥 𝑧 ≤ 𝑦 | |
10 | simp3 1135 | . . . . . . . . 9 ⊢ ((∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑦 ∧ 𝑥 ∈ 𝐴 ∧ 𝑧 = 𝐵) → 𝑧 = 𝐵) | |
11 | rspa 3235 | . . . . . . . . . 10 ⊢ ((∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑦 ∧ 𝑥 ∈ 𝐴) → 𝐵 ≤ 𝑦) | |
12 | 11 | 3adant3 1129 | . . . . . . . . 9 ⊢ ((∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑦 ∧ 𝑥 ∈ 𝐴 ∧ 𝑧 = 𝐵) → 𝐵 ≤ 𝑦) |
13 | 10, 12 | eqbrtrd 5171 | . . . . . . . 8 ⊢ ((∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑦 ∧ 𝑥 ∈ 𝐴 ∧ 𝑧 = 𝐵) → 𝑧 ≤ 𝑦) |
14 | 13 | 3exp 1116 | . . . . . . 7 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑦 → (𝑥 ∈ 𝐴 → (𝑧 = 𝐵 → 𝑧 ≤ 𝑦))) |
15 | 14 | adantl 480 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑦) → (𝑥 ∈ 𝐴 → (𝑧 = 𝐵 → 𝑧 ≤ 𝑦))) |
16 | 8, 9, 15 | rexlimd 3253 | . . . . 5 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑦) → (∃𝑥 ∈ 𝐴 𝑧 = 𝐵 → 𝑧 ≤ 𝑦)) |
17 | 16 | imp 405 | . . . 4 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑦) ∧ ∃𝑥 ∈ 𝐴 𝑧 = 𝐵) → 𝑧 ≤ 𝑦) |
18 | 3, 17 | sylan2b 592 | . . 3 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑦) ∧ 𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)) → 𝑧 ≤ 𝑦) |
19 | 18 | ralrimiva 3135 | . 2 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑦) → ∀𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑧 ≤ 𝑦) |
20 | rnmptbddlem.b | . 2 ⊢ (𝜑 → ∃𝑦 ∈ ℝ ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑦) | |
21 | 19, 20 | reximddv3 3161 | 1 ⊢ (𝜑 → ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑧 ≤ 𝑦) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 ∧ w3a 1084 = wceq 1533 Ⅎwnf 1777 ∈ wcel 2098 ∀wral 3050 ∃wrex 3059 Vcvv 3461 class class class wbr 5149 ↦ cmpt 5232 ran crn 5679 ℝcr 11139 ≤ cle 11281 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5300 ax-nul 5307 ax-pr 5429 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ral 3051 df-rex 3060 df-rab 3419 df-v 3463 df-dif 3947 df-un 3949 df-ss 3961 df-nul 4323 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-br 5150 df-opab 5212 df-mpt 5233 df-cnv 5686 df-dm 5688 df-rn 5689 |
This theorem is referenced by: rnmptbdd 44759 |
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