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Mirrors > Home > MPE Home > Th. List > f00 | Structured version Visualization version 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 6490 | . . . . 5 ⊢ (𝐹:𝐴⟶∅ → Fun 𝐹) | |
2 | frn 6493 | . . . . . . 7 ⊢ (𝐹:𝐴⟶∅ → ran 𝐹 ⊆ ∅) | |
3 | ss0 4306 | . . . . . . 7 ⊢ (ran 𝐹 ⊆ ∅ → ran 𝐹 = ∅) | |
4 | 2, 3 | syl 17 | . . . . . 6 ⊢ (𝐹:𝐴⟶∅ → ran 𝐹 = ∅) |
5 | dm0rn0 5759 | . . . . . 6 ⊢ (dom 𝐹 = ∅ ↔ ran 𝐹 = ∅) | |
6 | 4, 5 | sylibr 237 | . . . . 5 ⊢ (𝐹:𝐴⟶∅ → dom 𝐹 = ∅) |
7 | df-fn 6327 | . . . . 5 ⊢ (𝐹 Fn ∅ ↔ (Fun 𝐹 ∧ dom 𝐹 = ∅)) | |
8 | 1, 6, 7 | sylanbrc 586 | . . . 4 ⊢ (𝐹:𝐴⟶∅ → 𝐹 Fn ∅) |
9 | fn0 6451 | . . . 4 ⊢ (𝐹 Fn ∅ ↔ 𝐹 = ∅) | |
10 | 8, 9 | sylib 221 | . . 3 ⊢ (𝐹:𝐴⟶∅ → 𝐹 = ∅) |
11 | fdm 6495 | . . . 4 ⊢ (𝐹:𝐴⟶∅ → dom 𝐹 = 𝐴) | |
12 | 11, 6 | eqtr3d 2835 | . . 3 ⊢ (𝐹:𝐴⟶∅ → 𝐴 = ∅) |
13 | 10, 12 | jca 515 | . 2 ⊢ (𝐹:𝐴⟶∅ → (𝐹 = ∅ ∧ 𝐴 = ∅)) |
14 | f0 6534 | . . 3 ⊢ ∅:∅⟶∅ | |
15 | feq1 6468 | . . . 4 ⊢ (𝐹 = ∅ → (𝐹:𝐴⟶∅ ↔ ∅:𝐴⟶∅)) | |
16 | feq2 6469 | . . . 4 ⊢ (𝐴 = ∅ → (∅:𝐴⟶∅ ↔ ∅:∅⟶∅)) | |
17 | 15, 16 | sylan9bb 513 | . . 3 ⊢ ((𝐹 = ∅ ∧ 𝐴 = ∅) → (𝐹:𝐴⟶∅ ↔ ∅:∅⟶∅)) |
18 | 14, 17 | mpbiri 261 | . 2 ⊢ ((𝐹 = ∅ ∧ 𝐴 = ∅) → 𝐹:𝐴⟶∅) |
19 | 13, 18 | impbii 212 | 1 ⊢ (𝐹:𝐴⟶∅ ↔ (𝐹 = ∅ ∧ 𝐴 = ∅)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 ∧ wa 399 = wceq 1538 ⊆ wss 3881 ∅c0 4243 dom cdm 5519 ran crn 5520 Fun wfun 6318 Fn wfn 6319 ⟶wf 6320 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-v 3443 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-br 5031 df-opab 5093 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-fun 6326 df-fn 6327 df-f 6328 |
This theorem is referenced by: cantnff 9121 0wrd0 13883 supcvg 15203 ram0 16348 itgsubstlem 24651 uhgr0vb 26865 lfuhgr1v0e 27044 wlkv0 27440 sate0fv0 32777 prv0 32790 ismgmOLD 35288 |
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