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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 3005 | . . 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 40219 | . . 3 ⊢ (𝜑 → (ran 𝐹 = ∅ ↔ 𝐴 = ∅)) |
7 | 2, 6 | mtbird 317 | . 2 ⊢ (𝜑 → ¬ ran 𝐹 = ∅) |
8 | 7 | neqned 3007 | 1 ⊢ (𝜑 → ran 𝐹 ≠ ∅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 = wceq 1658 Ⅎwnf 1884 ∈ wcel 2166 ≠ wne 3000 ∅c0 4145 ↦ cmpt 4953 ran crn 5344 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2804 ax-sep 5006 ax-nul 5014 ax-pr 5128 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2606 df-eu 2641 df-clab 2813 df-cleq 2819 df-clel 2822 df-nfc 2959 df-ne 3001 df-ral 3123 df-rab 3127 df-v 3417 df-dif 3802 df-un 3804 df-in 3806 df-ss 3813 df-nul 4146 df-if 4308 df-sn 4399 df-pr 4401 df-op 4405 df-br 4875 df-opab 4937 df-mpt 4954 df-xp 5349 df-rel 5350 df-cnv 5351 df-dm 5353 df-rn 5354 df-res 5355 df-ima 5356 |
This theorem is referenced by: infnsuprnmpt 40266 suprclrnmpt 40267 fisupclrnmpt 40418 supxrrernmpt 40444 suprleubrnmpt 40445 supxrre3rnmpt 40452 supminfrnmpt 40468 infrpgernmpt 40490 limsupvaluz2 40766 ioorrnopnlem 41316 iunhoiioolem 41684 vonioolem1 41689 |
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