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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 map-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 2787 | . . 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 38407 | . . 3 ⊢ (𝜑 → (ran 𝐹 = ∅ ↔ 𝐴 = ∅)) |
7 | 2, 6 | mtbird 314 | . 2 ⊢ (𝜑 → ¬ ran 𝐹 = ∅) |
8 | 7 | neqned 2789 | 1 ⊢ (𝜑 → ran 𝐹 ≠ ∅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 Ⅎwnf 1699 ∈ wcel 1977 ≠ wne 2780 ∅c0 3874 ↦ cmpt 4643 ran crn 5039 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-br 4584 df-opab 4644 df-mpt 4645 df-xp 5044 df-rel 5045 df-cnv 5046 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 |
This theorem is referenced by: ioorrnopnlem 39200 iunhoiioolem 39566 vonioolem1 39571 |
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