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Mirrors > Home > MPE Home > Th. List > rnmpt0f | Structured version Visualization version GIF version |
Description: The range of a function in maps-to notation is empty if and only if its domain is empty. (Contributed by Glauco Siliprandi, 8-Apr-2021.) |
Ref | Expression |
---|---|
rnmpt0f.1 | ⊢ Ⅎ𝑥𝜑 |
rnmpt0f.2 | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉) |
rnmpt0f.3 | ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) |
Ref | Expression |
---|---|
rnmpt0f | ⊢ (𝜑 → (ran 𝐹 = ∅ ↔ 𝐴 = ∅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rnmpt0f.1 | . . . . . 6 ⊢ Ⅎ𝑥𝜑 | |
2 | rnmpt0f.2 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉) | |
3 | 2 | ex 412 | . . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → 𝐵 ∈ 𝑉)) |
4 | 1, 3 | ralrimi 3246 | . . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉) |
5 | dmmptg 6232 | . . . . 5 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 → dom (𝑥 ∈ 𝐴 ↦ 𝐵) = 𝐴) | |
6 | 4, 5 | syl 17 | . . . 4 ⊢ (𝜑 → dom (𝑥 ∈ 𝐴 ↦ 𝐵) = 𝐴) |
7 | 6 | eqcomd 2730 | . . 3 ⊢ (𝜑 → 𝐴 = dom (𝑥 ∈ 𝐴 ↦ 𝐵)) |
8 | 7 | eqeq1d 2726 | . 2 ⊢ (𝜑 → (𝐴 = ∅ ↔ dom (𝑥 ∈ 𝐴 ↦ 𝐵) = ∅)) |
9 | dm0rn0 5915 | . . 3 ⊢ (dom (𝑥 ∈ 𝐴 ↦ 𝐵) = ∅ ↔ ran (𝑥 ∈ 𝐴 ↦ 𝐵) = ∅) | |
10 | 9 | a1i 11 | . 2 ⊢ (𝜑 → (dom (𝑥 ∈ 𝐴 ↦ 𝐵) = ∅ ↔ ran (𝑥 ∈ 𝐴 ↦ 𝐵) = ∅)) |
11 | rnmpt0f.3 | . . . . . 6 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) | |
12 | 11 | rneqi 5927 | . . . . 5 ⊢ ran 𝐹 = ran (𝑥 ∈ 𝐴 ↦ 𝐵) |
13 | 12 | a1i 11 | . . . 4 ⊢ (𝜑 → ran 𝐹 = ran (𝑥 ∈ 𝐴 ↦ 𝐵)) |
14 | 13 | eqcomd 2730 | . . 3 ⊢ (𝜑 → ran (𝑥 ∈ 𝐴 ↦ 𝐵) = ran 𝐹) |
15 | 14 | eqeq1d 2726 | . 2 ⊢ (𝜑 → (ran (𝑥 ∈ 𝐴 ↦ 𝐵) = ∅ ↔ ran 𝐹 = ∅)) |
16 | 8, 10, 15 | 3bitrrd 306 | 1 ⊢ (𝜑 → (ran 𝐹 = ∅ ↔ 𝐴 = ∅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1533 Ⅎwnf 1777 ∈ wcel 2098 ∀wral 3053 ∅c0 4315 ↦ cmpt 5222 dom cdm 5667 ran crn 5668 |
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 2163 ax-ext 2695 ax-sep 5290 ax-nul 5297 ax-pr 5418 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ral 3054 df-rab 3425 df-v 3468 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-nul 4316 df-if 4522 df-sn 4622 df-pr 4624 df-op 4628 df-br 5140 df-opab 5202 df-mpt 5223 df-xp 5673 df-rel 5674 df-cnv 5675 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 |
This theorem is referenced by: rnmptn0 6234 |
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