![]() |
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 6261 | . . . 4 ⊢ (𝑋 = ∅ → (𝐹:𝑋⟶𝑌 ↔ 𝐹:∅⟶𝑌)) | |
2 | f0bi 6326 | . . . . 5 ⊢ (𝐹:∅⟶𝑌 ↔ 𝐹 = ∅) | |
3 | 2 | biimpi 208 | . . . 4 ⊢ (𝐹:∅⟶𝑌 → 𝐹 = ∅) |
4 | 1, 3 | syl6bi 245 | . . 3 ⊢ (𝑋 = ∅ → (𝐹:𝑋⟶𝑌 → 𝐹 = ∅)) |
5 | 4 | com12 32 | . 2 ⊢ (𝐹:𝑋⟶𝑌 → (𝑋 = ∅ → 𝐹 = ∅)) |
6 | feq1 6260 | . . . 4 ⊢ (𝐹 = ∅ → (𝐹:𝑋⟶𝑌 ↔ ∅:𝑋⟶𝑌)) | |
7 | dm0 5572 | . . . . 5 ⊢ dom ∅ = ∅ | |
8 | fdm 6287 | . . . . 5 ⊢ (∅:𝑋⟶𝑌 → dom ∅ = 𝑋) | |
9 | 7, 8 | syl5reqr 2877 | . . . 4 ⊢ (∅:𝑋⟶𝑌 → 𝑋 = ∅) |
10 | 6, 9 | syl6bi 245 | . . 3 ⊢ (𝐹 = ∅ → (𝐹:𝑋⟶𝑌 → 𝑋 = ∅)) |
11 | 10 | com12 32 | . 2 ⊢ (𝐹:𝑋⟶𝑌 → (𝐹 = ∅ → 𝑋 = ∅)) |
12 | 5, 11 | impbid 204 | 1 ⊢ (𝐹:𝑋⟶𝑌 → (𝑋 = ∅ ↔ 𝐹 = ∅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 = wceq 1658 ∅c0 4145 dom cdm 5343 ⟶wf 6120 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2804 ax-sep 5006 ax-nul 5014 ax-pr 5128 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2606 df-eu 2641 df-clab 2813 df-cleq 2819 df-clel 2822 df-nfc 2959 df-ral 3123 df-rex 3124 df-rab 3127 df-v 3417 df-dif 3802 df-un 3804 df-in 3806 df-ss 3813 df-nul 4146 df-if 4308 df-sn 4399 df-pr 4401 df-op 4405 df-br 4875 df-opab 4937 df-id 5251 df-xp 5349 df-rel 5350 df-cnv 5351 df-co 5352 df-dm 5353 df-rn 5354 df-fun 6126 df-fn 6127 df-f 6128 |
This theorem is referenced by: swrdn0OLD 13718 pfxn0 13766 elfrlmbasn0 20470 mavmulsolcl 20726 |
Copyright terms: Public domain | W3C validator |