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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 6738 | . . . . 5 ⊢ (𝐹:𝐴⟶∅ → Fun 𝐹) | |
| 2 | frn 6742 | . . . . . . 7 ⊢ (𝐹:𝐴⟶∅ → ran 𝐹 ⊆ ∅) | |
| 3 | ss0 4401 | . . . . . . 7 ⊢ (ran 𝐹 ⊆ ∅ → ran 𝐹 = ∅) | |
| 4 | 2, 3 | syl 17 | . . . . . 6 ⊢ (𝐹:𝐴⟶∅ → ran 𝐹 = ∅) | 
| 5 | dm0rn0 5934 | . . . . . 6 ⊢ (dom 𝐹 = ∅ ↔ ran 𝐹 = ∅) | |
| 6 | 4, 5 | sylibr 234 | . . . . 5 ⊢ (𝐹:𝐴⟶∅ → dom 𝐹 = ∅) | 
| 7 | df-fn 6563 | . . . . 5 ⊢ (𝐹 Fn ∅ ↔ (Fun 𝐹 ∧ dom 𝐹 = ∅)) | |
| 8 | 1, 6, 7 | sylanbrc 583 | . . . 4 ⊢ (𝐹:𝐴⟶∅ → 𝐹 Fn ∅) | 
| 9 | fn0 6698 | . . . 4 ⊢ (𝐹 Fn ∅ ↔ 𝐹 = ∅) | |
| 10 | 8, 9 | sylib 218 | . . 3 ⊢ (𝐹:𝐴⟶∅ → 𝐹 = ∅) | 
| 11 | fdm 6744 | . . . 4 ⊢ (𝐹:𝐴⟶∅ → dom 𝐹 = 𝐴) | |
| 12 | 11, 6 | eqtr3d 2778 | . . 3 ⊢ (𝐹:𝐴⟶∅ → 𝐴 = ∅) | 
| 13 | 10, 12 | jca 511 | . 2 ⊢ (𝐹:𝐴⟶∅ → (𝐹 = ∅ ∧ 𝐴 = ∅)) | 
| 14 | f0 6788 | . . 3 ⊢ ∅:∅⟶∅ | |
| 15 | feq1 6715 | . . . 4 ⊢ (𝐹 = ∅ → (𝐹:𝐴⟶∅ ↔ ∅:𝐴⟶∅)) | |
| 16 | feq2 6716 | . . . 4 ⊢ (𝐴 = ∅ → (∅:𝐴⟶∅ ↔ ∅:∅⟶∅)) | |
| 17 | 15, 16 | sylan9bb 509 | . . 3 ⊢ ((𝐹 = ∅ ∧ 𝐴 = ∅) → (𝐹:𝐴⟶∅ ↔ ∅:∅⟶∅)) | 
| 18 | 14, 17 | mpbiri 258 | . 2 ⊢ ((𝐹 = ∅ ∧ 𝐴 = ∅) → 𝐹:𝐴⟶∅) | 
| 19 | 13, 18 | impbii 209 | 1 ⊢ (𝐹:𝐴⟶∅ ↔ (𝐹 = ∅ ∧ 𝐴 = ∅)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ↔ wb 206 ∧ wa 395 = wceq 1539 ⊆ wss 3950 ∅c0 4332 dom cdm 5684 ran crn 5685 Fun wfun 6554 Fn wfn 6555 ⟶wf 6556 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pr 5431 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-clab 2714 df-cleq 2728 df-clel 2815 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-br 5143 df-opab 5205 df-id 5577 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-fun 6562 df-fn 6563 df-f 6564 | 
| This theorem is referenced by: dom0 9143 cantnff 9715 0wrd0 14579 supcvg 15893 ram0 17061 itgsubstlem 26090 uhgr0vb 29090 lfuhgr1v0e 29272 wlkv0 29670 sate0fv0 35423 prv0 35436 ismgmOLD 37858 mof0 48752 mof0ALT 48754 mofeu 48762 fdomne0 48764 f002 48768 fullthinc 49124 | 
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