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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 6737 | . . 3 ⊢ (𝐹:∅⟶𝑋 → 𝐹 Fn ∅) | |
2 | fn0 6700 | . . 3 ⊢ (𝐹 Fn ∅ ↔ 𝐹 = ∅) | |
3 | 1, 2 | sylib 218 | . 2 ⊢ (𝐹:∅⟶𝑋 → 𝐹 = ∅) |
4 | f0 6790 | . . 3 ⊢ ∅:∅⟶𝑋 | |
5 | feq1 6717 | . . 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 1537 ∅c0 4339 Fn wfn 6558 ⟶wf 6559 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-clab 2713 df-cleq 2727 df-clel 2814 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-br 5149 df-opab 5211 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-fun 6565 df-fn 6566 df-f 6567 |
This theorem is referenced by: f0dom0 6793 mapdm0 8881 fset0 8893 0map0sn0 8924 griedg0ssusgr 29297 rgrusgrprc 29622 sticksstones11 42138 2ffzoeq 47277 f102g 48682 |
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