Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > rnmptn0 | Structured version Visualization version GIF version |
Description: The range of a function in maps-to notation is nonempty if the domain is nonempty. (Contributed by Glauco Siliprandi, 8-Apr-2021.) |
Ref | Expression |
---|---|
rnmptn0.x | ⊢ Ⅎ𝑥𝜑 |
rnmptn0.b | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉) |
rnmptn0.f | ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) |
rnmptn0.a | ⊢ (𝜑 → 𝐴 ≠ ∅) |
Ref | Expression |
---|---|
rnmptn0 | ⊢ (𝜑 → ran 𝐹 ≠ ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rnmptn0.a | . . . 4 ⊢ (𝜑 → 𝐴 ≠ ∅) | |
2 | 1 | neneqd 3018 | . . 3 ⊢ (𝜑 → ¬ 𝐴 = ∅) |
3 | rnmptn0.x | . . . 4 ⊢ Ⅎ𝑥𝜑 | |
4 | rnmptn0.b | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉) | |
5 | rnmptn0.f | . . . 4 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) | |
6 | 3, 4, 5 | rnmpt0 41359 | . . 3 ⊢ (𝜑 → (ran 𝐹 = ∅ ↔ 𝐴 = ∅)) |
7 | 2, 6 | mtbird 326 | . 2 ⊢ (𝜑 → ¬ ran 𝐹 = ∅) |
8 | 7 | neqned 3020 | 1 ⊢ (𝜑 → ran 𝐹 ≠ ∅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1528 Ⅎwnf 1775 ∈ wcel 2105 ≠ wne 3013 ∅c0 4288 ↦ cmpt 5137 ran crn 5549 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rab 3144 df-v 3494 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-br 5058 df-opab 5120 df-mpt 5138 df-xp 5554 df-rel 5555 df-cnv 5556 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 |
This theorem is referenced by: infnsuprnmpt 41398 suprclrnmpt 41399 fisupclrnmpt 41547 supxrrernmpt 41571 suprleubrnmpt 41572 supxrre3rnmpt 41579 supminfrnmpt 41595 infrpgernmpt 41617 limsupvaluz2 41895 ioorrnopnlem 42466 iunhoiioolem 42834 vonioolem1 42839 |
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