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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 6717 | . . . 4 ⊢ (𝑋 = ∅ → (𝐹:𝑋⟶𝑌 ↔ 𝐹:∅⟶𝑌)) | |
| 2 | f0bi 6791 | . . . . 5 ⊢ (𝐹:∅⟶𝑌 ↔ 𝐹 = ∅) | |
| 3 | 2 | biimpi 216 | . . . 4 ⊢ (𝐹:∅⟶𝑌 → 𝐹 = ∅) |
| 4 | 1, 3 | biimtrdi 253 | . . 3 ⊢ (𝑋 = ∅ → (𝐹:𝑋⟶𝑌 → 𝐹 = ∅)) |
| 5 | 4 | com12 32 | . 2 ⊢ (𝐹:𝑋⟶𝑌 → (𝑋 = ∅ → 𝐹 = ∅)) |
| 6 | feq1 6716 | . . . 4 ⊢ (𝐹 = ∅ → (𝐹:𝑋⟶𝑌 ↔ ∅:𝑋⟶𝑌)) | |
| 7 | fdm 6745 | . . . . 5 ⊢ (∅:𝑋⟶𝑌 → dom ∅ = 𝑋) | |
| 8 | dm0 5931 | . . . . 5 ⊢ dom ∅ = ∅ | |
| 9 | 7, 8 | eqtr3di 2792 | . . . 4 ⊢ (∅:𝑋⟶𝑌 → 𝑋 = ∅) |
| 10 | 6, 9 | biimtrdi 253 | . . 3 ⊢ (𝐹 = ∅ → (𝐹:𝑋⟶𝑌 → 𝑋 = ∅)) |
| 11 | 10 | com12 32 | . 2 ⊢ (𝐹:𝑋⟶𝑌 → (𝐹 = ∅ → 𝑋 = ∅)) |
| 12 | 5, 11 | impbid 212 | 1 ⊢ (𝐹:𝑋⟶𝑌 → (𝑋 = ∅ ↔ 𝐹 = ∅)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 = wceq 1540 ∅c0 4333 dom cdm 5685 ⟶wf 6557 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-opab 5206 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-fun 6563 df-fn 6564 df-f 6565 |
| This theorem is referenced by: pfxn0 14724 elfrlmbasn0 21783 mavmulsolcl 22557 wrdpmtrlast 33113 fdomne0 48759 |
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