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| 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 6670 | . . 3 ⊢ (𝐹 Fn ∅ → Rel 𝐹) | |
| 2 | fndm 6671 | . . 3 ⊢ (𝐹 Fn ∅ → dom 𝐹 = ∅) | |
| 3 | reldm0 5938 | . . . 4 ⊢ (Rel 𝐹 → (𝐹 = ∅ ↔ dom 𝐹 = ∅)) | |
| 4 | 3 | biimpar 477 | . . 3 ⊢ ((Rel 𝐹 ∧ dom 𝐹 = ∅) → 𝐹 = ∅) | 
| 5 | 1, 2, 4 | syl2anc 584 | . 2 ⊢ (𝐹 Fn ∅ → 𝐹 = ∅) | 
| 6 | fun0 6631 | . . . 4 ⊢ Fun ∅ | |
| 7 | dm0 5931 | . . . 4 ⊢ dom ∅ = ∅ | |
| 8 | df-fn 6564 | . . . 4 ⊢ (∅ Fn ∅ ↔ (Fun ∅ ∧ dom ∅ = ∅)) | |
| 9 | 6, 7, 8 | mpbir2an 711 | . . 3 ⊢ ∅ Fn ∅ | 
| 10 | fneq1 6659 | . . 3 ⊢ (𝐹 = ∅ → (𝐹 Fn ∅ ↔ ∅ Fn ∅)) | |
| 11 | 9, 10 | mpbiri 258 | . 2 ⊢ (𝐹 = ∅ → 𝐹 Fn ∅) | 
| 12 | 5, 11 | impbii 209 | 1 ⊢ (𝐹 Fn ∅ ↔ 𝐹 = ∅) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ↔ wb 206 = wceq 1540 ∅c0 4333 dom cdm 5685 Rel wrel 5690 Fun wfun 6555 Fn wfn 6556 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-mo 2540 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-opab 5206 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-fun 6563 df-fn 6564 | 
| This theorem is referenced by: mpt0 6710 f0 6789 f00 6790 f0bi 6791 f1o00 6883 fo00 6884 tpos0 8281 ixp0x 8966 0fz1 13584 hashf1 14496 fuchom 18009 grpinvfvi 19000 mulgfval 19087 mulgfvalALT 19088 mulgfvi 19091 0frgp 19797 invrfval 20389 psrvscafval 21968 tmdgsum 24103 deg1fvi 26124 hon0 31812 fnchoice 45034 dvnprodlem3 45963 0funcg2 48917 0funcALT 48921 | 
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