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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 2992 | . . 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 41849 | . . 3 ⊢ (𝜑 → (ran 𝐹 = ∅ ↔ 𝐴 = ∅)) |
7 | 2, 6 | mtbird 328 | . 2 ⊢ (𝜑 → ¬ ran 𝐹 = ∅) |
8 | 7 | neqned 2994 | 1 ⊢ (𝜑 → ran 𝐹 ≠ ∅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 Ⅎwnf 1785 ∈ wcel 2111 ≠ wne 2987 ∅c0 4243 ↦ cmpt 5110 ran crn 5520 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rab 3115 df-v 3443 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-br 5031 df-opab 5093 df-mpt 5111 df-xp 5525 df-rel 5526 df-cnv 5527 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 |
This theorem is referenced by: infnsuprnmpt 41888 suprclrnmpt 41889 fisupclrnmpt 42035 supxrrernmpt 42058 suprleubrnmpt 42059 supxrre3rnmpt 42066 supminfrnmpt 42082 infrpgernmpt 42104 limsupvaluz2 42380 ioorrnopnlem 42946 iunhoiioolem 43314 vonioolem1 43319 |
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