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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 6704 | . . . 4 ⊢ (𝑋 = ∅ → (𝐹:𝑋⟶𝑌 ↔ 𝐹:∅⟶𝑌)) | |
2 | f0bi 6780 | . . . . 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 6703 | . . . 4 ⊢ (𝐹 = ∅ → (𝐹:𝑋⟶𝑌 ↔ ∅:𝑋⟶𝑌)) | |
7 | fdm 6731 | . . . . 5 ⊢ (∅:𝑋⟶𝑌 → dom ∅ = 𝑋) | |
8 | dm0 5923 | . . . . 5 ⊢ dom ∅ = ∅ | |
9 | 7, 8 | eqtr3di 2783 | . . . 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 1534 ∅c0 4323 dom cdm 5678 ⟶wf 6544 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pr 5429 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-clab 2706 df-cleq 2720 df-clel 2806 df-ral 3059 df-rex 3068 df-rab 3430 df-v 3473 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-br 5149 df-opab 5211 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-fun 6550 df-fn 6551 df-f 6552 |
This theorem is referenced by: pfxn0 14669 elfrlmbasn0 21697 mavmulsolcl 22466 fdomne0 47902 |
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