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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 5333 | . . 3 ⊢ (𝐹 Fn ∅ → Rel 𝐹) | |
2 | fndm 5334 | . . 3 ⊢ (𝐹 Fn ∅ → dom 𝐹 = ∅) | |
3 | reldm0 4863 | . . . 4 ⊢ (Rel 𝐹 → (𝐹 = ∅ ↔ dom 𝐹 = ∅)) | |
4 | 3 | biimpar 297 | . . 3 ⊢ ((Rel 𝐹 ∧ dom 𝐹 = ∅) → 𝐹 = ∅) |
5 | 1, 2, 4 | syl2anc 411 | . 2 ⊢ (𝐹 Fn ∅ → 𝐹 = ∅) |
6 | fun0 5293 | . . . 4 ⊢ Fun ∅ | |
7 | dm0 4859 | . . . 4 ⊢ dom ∅ = ∅ | |
8 | df-fn 5238 | . . . 4 ⊢ (∅ Fn ∅ ↔ (Fun ∅ ∧ dom ∅ = ∅)) | |
9 | 6, 7, 8 | mpbir2an 944 | . . 3 ⊢ ∅ Fn ∅ |
10 | fneq1 5323 | . . 3 ⊢ (𝐹 = ∅ → (𝐹 Fn ∅ ↔ ∅ Fn ∅)) | |
11 | 9, 10 | mpbiri 168 | . 2 ⊢ (𝐹 = ∅ → 𝐹 Fn ∅) |
12 | 5, 11 | impbii 126 | 1 ⊢ (𝐹 Fn ∅ ↔ 𝐹 = ∅) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 105 = wceq 1364 ∅c0 3437 dom cdm 4644 Rel wrel 4649 Fun wfun 5229 Fn wfn 5230 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-14 2163 ax-ext 2171 ax-sep 4136 ax-nul 4144 ax-pow 4192 ax-pr 4227 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ral 2473 df-rex 2474 df-v 2754 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-nul 3438 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-br 4019 df-opab 4080 df-id 4311 df-xp 4650 df-rel 4651 df-cnv 4652 df-co 4653 df-dm 4654 df-fun 5237 df-fn 5238 |
This theorem is referenced by: mpt0 5362 f0 5425 f00 5426 f0bi 5427 f1o00 5515 fo00 5516 tpos0 6300 ixp0x 6753 0fz1 10077 |
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