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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 5177 | . . 3 ⊢ (𝐹 Fn ∅ → Rel 𝐹) | |
2 | fndm 5178 | . . 3 ⊢ (𝐹 Fn ∅ → dom 𝐹 = ∅) | |
3 | reldm0 4715 | . . . 4 ⊢ (Rel 𝐹 → (𝐹 = ∅ ↔ dom 𝐹 = ∅)) | |
4 | 3 | biimpar 293 | . . 3 ⊢ ((Rel 𝐹 ∧ dom 𝐹 = ∅) → 𝐹 = ∅) |
5 | 1, 2, 4 | syl2anc 406 | . 2 ⊢ (𝐹 Fn ∅ → 𝐹 = ∅) |
6 | fun0 5137 | . . . 4 ⊢ Fun ∅ | |
7 | dm0 4711 | . . . 4 ⊢ dom ∅ = ∅ | |
8 | df-fn 5082 | . . . 4 ⊢ (∅ Fn ∅ ↔ (Fun ∅ ∧ dom ∅ = ∅)) | |
9 | 6, 7, 8 | mpbir2an 907 | . . 3 ⊢ ∅ Fn ∅ |
10 | fneq1 5167 | . . 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 1312 ∅c0 3327 dom cdm 4497 Rel wrel 4502 Fun wfun 5073 Fn wfn 5074 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 586 ax-in2 587 ax-io 681 ax-5 1404 ax-7 1405 ax-gen 1406 ax-ie1 1450 ax-ie2 1451 ax-8 1463 ax-10 1464 ax-11 1465 ax-i12 1466 ax-bndl 1467 ax-4 1468 ax-14 1473 ax-17 1487 ax-i9 1491 ax-ial 1495 ax-i5r 1496 ax-ext 2095 ax-sep 4004 ax-nul 4012 ax-pow 4056 ax-pr 4089 |
This theorem depends on definitions: df-bi 116 df-3an 945 df-tru 1315 df-fal 1318 df-nf 1418 df-sb 1717 df-eu 1976 df-mo 1977 df-clab 2100 df-cleq 2106 df-clel 2109 df-nfc 2242 df-ral 2393 df-rex 2394 df-v 2657 df-dif 3037 df-un 3039 df-in 3041 df-ss 3048 df-nul 3328 df-pw 3476 df-sn 3497 df-pr 3498 df-op 3500 df-br 3894 df-opab 3948 df-id 4173 df-xp 4503 df-rel 4504 df-cnv 4505 df-co 4506 df-dm 4507 df-fun 5081 df-fn 5082 |
This theorem is referenced by: mpt0 5206 f0 5269 f00 5270 f0bi 5271 f1o00 5356 fo00 5357 tpos0 6123 ixp0x 6572 0fz1 9712 |
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