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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 6507 | . . 3 ⊢ (𝐹:∅⟶𝑋 → 𝐹 Fn ∅) | |
2 | fn0 6472 | . . 3 ⊢ (𝐹 Fn ∅ ↔ 𝐹 = ∅) | |
3 | 1, 2 | sylib 219 | . 2 ⊢ (𝐹:∅⟶𝑋 → 𝐹 = ∅) |
4 | f0 6553 | . . 3 ⊢ ∅:∅⟶𝑋 | |
5 | feq1 6488 | . . 3 ⊢ (𝐹 = ∅ → (𝐹:∅⟶𝑋 ↔ ∅:∅⟶𝑋)) | |
6 | 4, 5 | mpbiri 259 | . 2 ⊢ (𝐹 = ∅ → 𝐹:∅⟶𝑋) |
7 | 3, 6 | impbii 210 | 1 ⊢ (𝐹:∅⟶𝑋 ↔ 𝐹 = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 207 = wceq 1528 ∅c0 4288 Fn wfn 6343 ⟶wf 6344 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-br 5058 df-opab 5120 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-fun 6350 df-fn 6351 df-f 6352 |
This theorem is referenced by: f0dom0 6556 mapdm0 8410 griedg0ssusgr 26974 rgrusgrprc 27298 2ffzoeq 43405 efmndbas0 43988 |
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