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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 416 | . . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → 𝐵 ∈ 𝑉)) |
4 | 1, 3 | ralrimi 3127 | . . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉) |
5 | dmmptg 6085 | . . . . 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 2742 | . . 3 ⊢ (𝜑 → 𝐴 = dom (𝑥 ∈ 𝐴 ↦ 𝐵)) |
8 | 7 | eqeq1d 2738 | . 2 ⊢ (𝜑 → (𝐴 = ∅ ↔ dom (𝑥 ∈ 𝐴 ↦ 𝐵) = ∅)) |
9 | dm0rn0 5779 | . . 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 5791 | . . . . 5 ⊢ ran 𝐹 = ran (𝑥 ∈ 𝐴 ↦ 𝐵) |
13 | 12 | a1i 11 | . . . 4 ⊢ (𝜑 → ran 𝐹 = ran (𝑥 ∈ 𝐴 ↦ 𝐵)) |
14 | 13 | eqcomd 2742 | . . 3 ⊢ (𝜑 → ran (𝑥 ∈ 𝐴 ↦ 𝐵) = ran 𝐹) |
15 | 14 | eqeq1d 2738 | . 2 ⊢ (𝜑 → (ran (𝑥 ∈ 𝐴 ↦ 𝐵) = ∅ ↔ ran 𝐹 = ∅)) |
16 | 8, 10, 15 | 3bitrrd 309 | 1 ⊢ (𝜑 → (ran 𝐹 = ∅ ↔ 𝐴 = ∅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1543 Ⅎwnf 1791 ∈ wcel 2112 ∀wral 3051 ∅c0 4223 ↦ cmpt 5120 dom cdm 5536 ran crn 5537 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pr 5307 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ral 3056 df-rab 3060 df-v 3400 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-sn 4528 df-pr 4530 df-op 4534 df-br 5040 df-opab 5102 df-mpt 5121 df-xp 5542 df-rel 5543 df-cnv 5544 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 |
This theorem is referenced by: rnmptn0 6087 |
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