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| Mirrors > Home > MPE Home > Th. List > f0bi | Structured version Visualization version GIF version | ||
| Description: A function with empty domain is empty. (Contributed by Alexander van der Vekens, 30-Jun-2018.) | 
| Ref | Expression | 
|---|---|
| f0bi | ⊢ (𝐹:∅⟶𝑋 ↔ 𝐹 = ∅) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | ffn 6736 | . . 3 ⊢ (𝐹:∅⟶𝑋 → 𝐹 Fn ∅) | |
| 2 | fn0 6699 | . . 3 ⊢ (𝐹 Fn ∅ ↔ 𝐹 = ∅) | |
| 3 | 1, 2 | sylib 218 | . 2 ⊢ (𝐹:∅⟶𝑋 → 𝐹 = ∅) | 
| 4 | f0 6789 | . . 3 ⊢ ∅:∅⟶𝑋 | |
| 5 | feq1 6716 | . . 3 ⊢ (𝐹 = ∅ → (𝐹:∅⟶𝑋 ↔ ∅:∅⟶𝑋)) | |
| 6 | 4, 5 | mpbiri 258 | . 2 ⊢ (𝐹 = ∅ → 𝐹:∅⟶𝑋) | 
| 7 | 3, 6 | impbii 209 | 1 ⊢ (𝐹:∅⟶𝑋 ↔ 𝐹 = ∅) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ↔ wb 206 = wceq 1540 ∅c0 4333 Fn wfn 6556 ⟶wf 6557 | 
| 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-10 2141 ax-11 2157 ax-12 2177 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-nf 1784 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-rn 5696 df-fun 6563 df-fn 6564 df-f 6565 | 
| This theorem is referenced by: f0dom0 6792 mapdm0 8882 fset0 8894 0map0sn0 8925 griedg0ssusgr 29282 rgrusgrprc 29607 sticksstones11 42157 2ffzoeq 47339 f102g 48761 0funcg2 48917 0funcALT 48921 | 
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