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Mirrors > Home > ILE Home > Th. List > f0dom0 | 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 5256 | . . . 4 ⊢ (𝑋 = ∅ → (𝐹:𝑋⟶𝑌 ↔ 𝐹:∅⟶𝑌)) | |
2 | f0bi 5315 | . . . . 5 ⊢ (𝐹:∅⟶𝑌 ↔ 𝐹 = ∅) | |
3 | 2 | biimpi 119 | . . . 4 ⊢ (𝐹:∅⟶𝑌 → 𝐹 = ∅) |
4 | 1, 3 | syl6bi 162 | . . 3 ⊢ (𝑋 = ∅ → (𝐹:𝑋⟶𝑌 → 𝐹 = ∅)) |
5 | 4 | com12 30 | . 2 ⊢ (𝐹:𝑋⟶𝑌 → (𝑋 = ∅ → 𝐹 = ∅)) |
6 | feq1 5255 | . . . 4 ⊢ (𝐹 = ∅ → (𝐹:𝑋⟶𝑌 ↔ ∅:𝑋⟶𝑌)) | |
7 | dm0 4753 | . . . . 5 ⊢ dom ∅ = ∅ | |
8 | fdm 5278 | . . . . 5 ⊢ (∅:𝑋⟶𝑌 → dom ∅ = 𝑋) | |
9 | 7, 8 | syl5reqr 2187 | . . . 4 ⊢ (∅:𝑋⟶𝑌 → 𝑋 = ∅) |
10 | 6, 9 | syl6bi 162 | . . 3 ⊢ (𝐹 = ∅ → (𝐹:𝑋⟶𝑌 → 𝑋 = ∅)) |
11 | 10 | com12 30 | . 2 ⊢ (𝐹:𝑋⟶𝑌 → (𝐹 = ∅ → 𝑋 = ∅)) |
12 | 5, 11 | impbid 128 | 1 ⊢ (𝐹:𝑋⟶𝑌 → (𝑋 = ∅ ↔ 𝐹 = ∅)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 = wceq 1331 ∅c0 3363 dom cdm 4539 ⟶wf 5119 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-nul 4054 ax-pow 4098 ax-pr 4131 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-v 2688 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-br 3930 df-opab 3990 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-fun 5125 df-fn 5126 df-f 5127 |
This theorem is referenced by: (None) |
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