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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 5314 | . . 3 ⊢ (𝐹 Fn ∅ → Rel 𝐹) | |
2 | fndm 5315 | . . 3 ⊢ (𝐹 Fn ∅ → dom 𝐹 = ∅) | |
3 | reldm0 4845 | . . . 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 5274 | . . . 4 ⊢ Fun ∅ | |
7 | dm0 4841 | . . . 4 ⊢ dom ∅ = ∅ | |
8 | df-fn 5219 | . . . 4 ⊢ (∅ Fn ∅ ↔ (Fun ∅ ∧ dom ∅ = ∅)) | |
9 | 6, 7, 8 | mpbir2an 942 | . . 3 ⊢ ∅ Fn ∅ |
10 | fneq1 5304 | . . 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 1353 ∅c0 3422 dom cdm 4626 Rel wrel 4631 Fun wfun 5210 Fn wfn 5211 |
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 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-14 2151 ax-ext 2159 ax-sep 4121 ax-nul 4129 ax-pow 4174 ax-pr 4209 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-rex 2461 df-v 2739 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-nul 3423 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-br 4004 df-opab 4065 df-id 4293 df-xp 4632 df-rel 4633 df-cnv 4634 df-co 4635 df-dm 4636 df-fun 5218 df-fn 5219 |
This theorem is referenced by: mpt0 5343 f0 5406 f00 5407 f0bi 5408 f1o00 5496 fo00 5497 tpos0 6274 ixp0x 6725 0fz1 10042 |
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