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Mirrors > Home > ILE Home > Th. List > f00 | GIF version |
Description: A class is a function with empty codomain iff it and its domain are empty. (Contributed by NM, 10-Dec-2003.) |
Ref | Expression |
---|---|
f00 | ⊢ (𝐹:𝐴⟶∅ ↔ (𝐹 = ∅ ∧ 𝐴 = ∅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ffun 5068 | . . . . 5 ⊢ (𝐹:𝐴⟶∅ → Fun 𝐹) | |
2 | frn 5072 | . . . . . . 7 ⊢ (𝐹:𝐴⟶∅ → ran 𝐹 ⊆ ∅) | |
3 | ss0 3284 | . . . . . . 7 ⊢ (ran 𝐹 ⊆ ∅ → ran 𝐹 = ∅) | |
4 | 2, 3 | syl 14 | . . . . . 6 ⊢ (𝐹:𝐴⟶∅ → ran 𝐹 = ∅) |
5 | dm0rn0 4570 | . . . . . 6 ⊢ (dom 𝐹 = ∅ ↔ ran 𝐹 = ∅) | |
6 | 4, 5 | sylibr 132 | . . . . 5 ⊢ (𝐹:𝐴⟶∅ → dom 𝐹 = ∅) |
7 | df-fn 4925 | . . . . 5 ⊢ (𝐹 Fn ∅ ↔ (Fun 𝐹 ∧ dom 𝐹 = ∅)) | |
8 | 1, 6, 7 | sylanbrc 408 | . . . 4 ⊢ (𝐹:𝐴⟶∅ → 𝐹 Fn ∅) |
9 | fn0 5038 | . . . 4 ⊢ (𝐹 Fn ∅ ↔ 𝐹 = ∅) | |
10 | 8, 9 | sylib 120 | . . 3 ⊢ (𝐹:𝐴⟶∅ → 𝐹 = ∅) |
11 | fdm 5070 | . . . 4 ⊢ (𝐹:𝐴⟶∅ → dom 𝐹 = 𝐴) | |
12 | 11, 6 | eqtr3d 2115 | . . 3 ⊢ (𝐹:𝐴⟶∅ → 𝐴 = ∅) |
13 | 10, 12 | jca 300 | . 2 ⊢ (𝐹:𝐴⟶∅ → (𝐹 = ∅ ∧ 𝐴 = ∅)) |
14 | f0 5100 | . . 3 ⊢ ∅:∅⟶∅ | |
15 | feq1 5050 | . . . 4 ⊢ (𝐹 = ∅ → (𝐹:𝐴⟶∅ ↔ ∅:𝐴⟶∅)) | |
16 | feq2 5051 | . . . 4 ⊢ (𝐴 = ∅ → (∅:𝐴⟶∅ ↔ ∅:∅⟶∅)) | |
17 | 15, 16 | sylan9bb 449 | . . 3 ⊢ ((𝐹 = ∅ ∧ 𝐴 = ∅) → (𝐹:𝐴⟶∅ ↔ ∅:∅⟶∅)) |
18 | 14, 17 | mpbiri 166 | . 2 ⊢ ((𝐹 = ∅ ∧ 𝐴 = ∅) → 𝐹:𝐴⟶∅) |
19 | 13, 18 | impbii 124 | 1 ⊢ (𝐹:𝐴⟶∅ ↔ (𝐹 = ∅ ∧ 𝐴 = ∅)) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 102 ↔ wb 103 = wceq 1284 ⊆ wss 2973 ∅c0 3251 dom cdm 4363 ran crn 4364 Fun wfun 4916 Fn wfn 4917 ⟶wf 4918 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-nul 3904 ax-pow 3948 ax-pr 3964 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-fal 1290 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ral 2353 df-rex 2354 df-v 2603 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-nul 3252 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-br 3786 df-opab 3840 df-id 4048 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-rn 4374 df-fun 4924 df-fn 4925 df-f 4926 |
This theorem is referenced by: (None) |
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