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 3021 | . . 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 41476 | . . 3 ⊢ (𝜑 → (ran 𝐹 = ∅ ↔ 𝐴 = ∅)) |
7 | 2, 6 | mtbird 327 | . 2 ⊢ (𝜑 → ¬ ran 𝐹 = ∅) |
8 | 7 | neqned 3023 | 1 ⊢ (𝜑 → ran 𝐹 ≠ ∅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 Ⅎwnf 1780 ∈ wcel 2110 ≠ wne 3016 ∅c0 4290 ↦ cmpt 5138 ran crn 5550 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pr 5321 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rab 3147 df-v 3496 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4561 df-pr 4563 df-op 4567 df-br 5059 df-opab 5121 df-mpt 5139 df-xp 5555 df-rel 5556 df-cnv 5557 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 |
This theorem is referenced by: infnsuprnmpt 41515 suprclrnmpt 41516 fisupclrnmpt 41664 supxrrernmpt 41688 suprleubrnmpt 41689 supxrre3rnmpt 41696 supminfrnmpt 41712 infrpgernmpt 41734 limsupvaluz2 42012 ioorrnopnlem 42583 iunhoiioolem 42951 vonioolem1 42956 |
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