![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > f0dom0 | Structured version Visualization version GIF version |
Description: A function is empty iff it has an empty domain. (Contributed by AV, 10-Feb-2019.) |
Ref | Expression |
---|---|
f0dom0 | ⊢ (𝐹:𝑋⟶𝑌 → (𝑋 = ∅ ↔ 𝐹 = ∅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | feq2 6692 | . . . 4 ⊢ (𝑋 = ∅ → (𝐹:𝑋⟶𝑌 ↔ 𝐹:∅⟶𝑌)) | |
2 | f0bi 6767 | . . . . 5 ⊢ (𝐹:∅⟶𝑌 ↔ 𝐹 = ∅) | |
3 | 2 | biimpi 215 | . . . 4 ⊢ (𝐹:∅⟶𝑌 → 𝐹 = ∅) |
4 | 1, 3 | biimtrdi 252 | . . 3 ⊢ (𝑋 = ∅ → (𝐹:𝑋⟶𝑌 → 𝐹 = ∅)) |
5 | 4 | com12 32 | . 2 ⊢ (𝐹:𝑋⟶𝑌 → (𝑋 = ∅ → 𝐹 = ∅)) |
6 | feq1 6691 | . . . 4 ⊢ (𝐹 = ∅ → (𝐹:𝑋⟶𝑌 ↔ ∅:𝑋⟶𝑌)) | |
7 | fdm 6719 | . . . . 5 ⊢ (∅:𝑋⟶𝑌 → dom ∅ = 𝑋) | |
8 | dm0 5913 | . . . . 5 ⊢ dom ∅ = ∅ | |
9 | 7, 8 | eqtr3di 2781 | . . . 4 ⊢ (∅:𝑋⟶𝑌 → 𝑋 = ∅) |
10 | 6, 9 | biimtrdi 252 | . . 3 ⊢ (𝐹 = ∅ → (𝐹:𝑋⟶𝑌 → 𝑋 = ∅)) |
11 | 10 | com12 32 | . 2 ⊢ (𝐹:𝑋⟶𝑌 → (𝐹 = ∅ → 𝑋 = ∅)) |
12 | 5, 11 | impbid 211 | 1 ⊢ (𝐹:𝑋⟶𝑌 → (𝑋 = ∅ ↔ 𝐹 = ∅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 = wceq 1533 ∅c0 4317 dom cdm 5669 ⟶wf 6532 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pr 5420 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-clab 2704 df-cleq 2718 df-clel 2804 df-ral 3056 df-rex 3065 df-rab 3427 df-v 3470 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-br 5142 df-opab 5204 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-fun 6538 df-fn 6539 df-f 6540 |
This theorem is referenced by: pfxn0 14640 elfrlmbasn0 21654 mavmulsolcl 22404 fdomne0 47771 |
Copyright terms: Public domain | W3C validator |