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Mirrors > Home > ILE Home > Th. List > fn0 | GIF version |
Description: A function with empty domain is empty. (Contributed by NM, 15-Apr-1998.) (Proof shortened by Andrew Salmon, 17-Sep-2011.) |
Ref | Expression |
---|---|
fn0 | ⊢ (𝐹 Fn ∅ ↔ 𝐹 = ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fnrel 5191 | . . 3 ⊢ (𝐹 Fn ∅ → Rel 𝐹) | |
2 | fndm 5192 | . . 3 ⊢ (𝐹 Fn ∅ → dom 𝐹 = ∅) | |
3 | reldm0 4727 | . . . 4 ⊢ (Rel 𝐹 → (𝐹 = ∅ ↔ dom 𝐹 = ∅)) | |
4 | 3 | biimpar 295 | . . 3 ⊢ ((Rel 𝐹 ∧ dom 𝐹 = ∅) → 𝐹 = ∅) |
5 | 1, 2, 4 | syl2anc 408 | . 2 ⊢ (𝐹 Fn ∅ → 𝐹 = ∅) |
6 | fun0 5151 | . . . 4 ⊢ Fun ∅ | |
7 | dm0 4723 | . . . 4 ⊢ dom ∅ = ∅ | |
8 | df-fn 5096 | . . . 4 ⊢ (∅ Fn ∅ ↔ (Fun ∅ ∧ dom ∅ = ∅)) | |
9 | 6, 7, 8 | mpbir2an 911 | . . 3 ⊢ ∅ Fn ∅ |
10 | fneq1 5181 | . . 3 ⊢ (𝐹 = ∅ → (𝐹 Fn ∅ ↔ ∅ Fn ∅)) | |
11 | 9, 10 | mpbiri 167 | . 2 ⊢ (𝐹 = ∅ → 𝐹 Fn ∅) |
12 | 5, 11 | impbii 125 | 1 ⊢ (𝐹 Fn ∅ ↔ 𝐹 = ∅) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 104 = wceq 1316 ∅c0 3333 dom cdm 4509 Rel wrel 4514 Fun wfun 5087 Fn wfn 5088 |
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 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-nul 4024 ax-pow 4068 ax-pr 4101 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-fal 1322 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ral 2398 df-rex 2399 df-v 2662 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-nul 3334 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-br 3900 df-opab 3960 df-id 4185 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-fun 5095 df-fn 5096 |
This theorem is referenced by: mpt0 5220 f0 5283 f00 5284 f0bi 5285 f1o00 5370 fo00 5371 tpos0 6139 ixp0x 6588 0fz1 9793 |
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