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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 5961 | . . . . 5 ⊢ (𝐹:𝐴⟶∅ → Fun 𝐹) | |
2 | frn 5966 | . . . . . . 7 ⊢ (𝐹:𝐴⟶∅ → ran 𝐹 ⊆ ∅) | |
3 | ss0 3926 | . . . . . . 7 ⊢ (ran 𝐹 ⊆ ∅ → ran 𝐹 = ∅) | |
4 | 2, 3 | syl 17 | . . . . . 6 ⊢ (𝐹:𝐴⟶∅ → ran 𝐹 = ∅) |
5 | dm0rn0 5263 | . . . . . 6 ⊢ (dom 𝐹 = ∅ ↔ ran 𝐹 = ∅) | |
6 | 4, 5 | sylibr 223 | . . . . 5 ⊢ (𝐹:𝐴⟶∅ → dom 𝐹 = ∅) |
7 | df-fn 5807 | . . . . 5 ⊢ (𝐹 Fn ∅ ↔ (Fun 𝐹 ∧ dom 𝐹 = ∅)) | |
8 | 1, 6, 7 | sylanbrc 695 | . . . 4 ⊢ (𝐹:𝐴⟶∅ → 𝐹 Fn ∅) |
9 | fn0 5924 | . . . 4 ⊢ (𝐹 Fn ∅ ↔ 𝐹 = ∅) | |
10 | 8, 9 | sylib 207 | . . 3 ⊢ (𝐹:𝐴⟶∅ → 𝐹 = ∅) |
11 | fdm 5964 | . . . 4 ⊢ (𝐹:𝐴⟶∅ → dom 𝐹 = 𝐴) | |
12 | 11, 6 | eqtr3d 2646 | . . 3 ⊢ (𝐹:𝐴⟶∅ → 𝐴 = ∅) |
13 | 10, 12 | jca 553 | . 2 ⊢ (𝐹:𝐴⟶∅ → (𝐹 = ∅ ∧ 𝐴 = ∅)) |
14 | f0 5999 | . . 3 ⊢ ∅:∅⟶∅ | |
15 | feq1 5939 | . . . 4 ⊢ (𝐹 = ∅ → (𝐹:𝐴⟶∅ ↔ ∅:𝐴⟶∅)) | |
16 | feq2 5940 | . . . 4 ⊢ (𝐴 = ∅ → (∅:𝐴⟶∅ ↔ ∅:∅⟶∅)) | |
17 | 15, 16 | sylan9bb 732 | . . 3 ⊢ ((𝐹 = ∅ ∧ 𝐴 = ∅) → (𝐹:𝐴⟶∅ ↔ ∅:∅⟶∅)) |
18 | 14, 17 | mpbiri 247 | . 2 ⊢ ((𝐹 = ∅ ∧ 𝐴 = ∅) → 𝐹:𝐴⟶∅) |
19 | 13, 18 | impbii 198 | 1 ⊢ (𝐹:𝐴⟶∅ ↔ (𝐹 = ∅ ∧ 𝐴 = ∅)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 195 ∧ wa 383 = wceq 1475 ⊆ wss 3540 ∅c0 3874 dom cdm 5038 ran crn 5039 Fun wfun 5798 Fn wfn 5799 ⟶wf 5800 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-br 4584 df-opab 4644 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-fun 5806 df-fn 5807 df-f 5808 |
This theorem is referenced by: cantnff 8454 0wrd0 13186 supcvg 14427 ram0 15564 itgsubstlem 23615 uhgr0vb 25738 uhgra0v 25839 usgra0v 25900 usgra1v 25919 wlkv0 26288 ismgmOLD 32819 lfuhgr1v0e 40480 1wlkv0 40859 |
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