Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > rnmpt0 | 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 |
---|---|
rnmpt0.1 | ⊢ Ⅎ𝑥𝜑 |
rnmpt0.2 | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉) |
rnmpt0.3 | ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) |
Ref | Expression |
---|---|
rnmpt0 | ⊢ (𝜑 → (ran 𝐹 = ∅ ↔ 𝐴 = ∅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rnmpt0.1 | . . . . . 6 ⊢ Ⅎ𝑥𝜑 | |
2 | rnmpt0.2 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉) | |
3 | 1, 2 | ralrimia 41418 | . . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉) |
4 | dmmptg 6096 | . . . . 5 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 → dom (𝑥 ∈ 𝐴 ↦ 𝐵) = 𝐴) | |
5 | 3, 4 | syl 17 | . . . 4 ⊢ (𝜑 → dom (𝑥 ∈ 𝐴 ↦ 𝐵) = 𝐴) |
6 | 5 | eqcomd 2827 | . . 3 ⊢ (𝜑 → 𝐴 = dom (𝑥 ∈ 𝐴 ↦ 𝐵)) |
7 | 6 | eqeq1d 2823 | . 2 ⊢ (𝜑 → (𝐴 = ∅ ↔ dom (𝑥 ∈ 𝐴 ↦ 𝐵) = ∅)) |
8 | dm0rn0 5795 | . . 3 ⊢ (dom (𝑥 ∈ 𝐴 ↦ 𝐵) = ∅ ↔ ran (𝑥 ∈ 𝐴 ↦ 𝐵) = ∅) | |
9 | 8 | a1i 11 | . 2 ⊢ (𝜑 → (dom (𝑥 ∈ 𝐴 ↦ 𝐵) = ∅ ↔ ran (𝑥 ∈ 𝐴 ↦ 𝐵) = ∅)) |
10 | rnmpt0.3 | . . . . . 6 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) | |
11 | 10 | rneqi 5807 | . . . . 5 ⊢ ran 𝐹 = ran (𝑥 ∈ 𝐴 ↦ 𝐵) |
12 | 11 | a1i 11 | . . . 4 ⊢ (𝜑 → ran 𝐹 = ran (𝑥 ∈ 𝐴 ↦ 𝐵)) |
13 | 12 | eqcomd 2827 | . . 3 ⊢ (𝜑 → ran (𝑥 ∈ 𝐴 ↦ 𝐵) = ran 𝐹) |
14 | 13 | eqeq1d 2823 | . 2 ⊢ (𝜑 → (ran (𝑥 ∈ 𝐴 ↦ 𝐵) = ∅ ↔ ran 𝐹 = ∅)) |
15 | 7, 9, 14 | 3bitrrd 308 | 1 ⊢ (𝜑 → (ran 𝐹 = ∅ ↔ 𝐴 = ∅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1537 Ⅎwnf 1784 ∈ wcel 2114 ∀wral 3138 ∅c0 4291 ↦ cmpt 5146 dom cdm 5555 ran crn 5556 |
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 2793 ax-sep 5203 ax-nul 5210 ax-pr 5330 |
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 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rab 3147 df-v 3496 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-br 5067 df-opab 5129 df-mpt 5147 df-xp 5561 df-rel 5562 df-cnv 5563 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 |
This theorem is referenced by: rnmptn0 41504 |
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