![]() |
Mathbox for Zhi Wang |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > fdomne0 | Structured version Visualization version GIF version |
Description: A function with non-empty domain is non-empty and has non-empty codomain. (Contributed by Zhi Wang, 1-Oct-2024.) |
Ref | Expression |
---|---|
fdomne0 | ⊢ ((𝐹:𝑋⟶𝑌 ∧ 𝑋 ≠ ∅) → (𝐹 ≠ ∅ ∧ 𝑌 ≠ ∅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | f0dom0 6793 | . . . 4 ⊢ (𝐹:𝑋⟶𝑌 → (𝑋 = ∅ ↔ 𝐹 = ∅)) | |
2 | 1 | necon3bid 2983 | . . 3 ⊢ (𝐹:𝑋⟶𝑌 → (𝑋 ≠ ∅ ↔ 𝐹 ≠ ∅)) |
3 | 2 | biimpa 476 | . 2 ⊢ ((𝐹:𝑋⟶𝑌 ∧ 𝑋 ≠ ∅) → 𝐹 ≠ ∅) |
4 | feq3 6719 | . . . . . 6 ⊢ (𝑌 = ∅ → (𝐹:𝑋⟶𝑌 ↔ 𝐹:𝑋⟶∅)) | |
5 | f00 6791 | . . . . . . 7 ⊢ (𝐹:𝑋⟶∅ ↔ (𝐹 = ∅ ∧ 𝑋 = ∅)) | |
6 | 5 | simprbi 496 | . . . . . 6 ⊢ (𝐹:𝑋⟶∅ → 𝑋 = ∅) |
7 | 4, 6 | biimtrdi 253 | . . . . 5 ⊢ (𝑌 = ∅ → (𝐹:𝑋⟶𝑌 → 𝑋 = ∅)) |
8 | nne 2942 | . . . . 5 ⊢ (¬ 𝑋 ≠ ∅ ↔ 𝑋 = ∅) | |
9 | 7, 8 | imbitrrdi 252 | . . . 4 ⊢ (𝑌 = ∅ → (𝐹:𝑋⟶𝑌 → ¬ 𝑋 ≠ ∅)) |
10 | imnan 399 | . . . 4 ⊢ ((𝐹:𝑋⟶𝑌 → ¬ 𝑋 ≠ ∅) ↔ ¬ (𝐹:𝑋⟶𝑌 ∧ 𝑋 ≠ ∅)) | |
11 | 9, 10 | sylib 218 | . . 3 ⊢ (𝑌 = ∅ → ¬ (𝐹:𝑋⟶𝑌 ∧ 𝑋 ≠ ∅)) |
12 | 11 | necon2ai 2968 | . 2 ⊢ ((𝐹:𝑋⟶𝑌 ∧ 𝑋 ≠ ∅) → 𝑌 ≠ ∅) |
13 | 3, 12 | jca 511 | 1 ⊢ ((𝐹:𝑋⟶𝑌 ∧ 𝑋 ≠ ∅) → (𝐹 ≠ ∅ ∧ 𝑌 ≠ ∅)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1537 ≠ wne 2938 ∅c0 4339 ⟶wf 6559 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-clab 2713 df-cleq 2727 df-clel 2814 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-br 5149 df-opab 5211 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-fun 6565 df-fn 6566 df-f 6567 |
This theorem is referenced by: fullthinc 48846 |
Copyright terms: Public domain | W3C validator |