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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 6516 | . . 3 ⊢ (𝐹:∅⟶𝑋 → 𝐹 Fn ∅) | |
2 | fn0 6481 | . . 3 ⊢ (𝐹 Fn ∅ ↔ 𝐹 = ∅) | |
3 | 1, 2 | sylib 220 | . 2 ⊢ (𝐹:∅⟶𝑋 → 𝐹 = ∅) |
4 | f0 6562 | . . 3 ⊢ ∅:∅⟶𝑋 | |
5 | feq1 6497 | . . 3 ⊢ (𝐹 = ∅ → (𝐹:∅⟶𝑋 ↔ ∅:∅⟶𝑋)) | |
6 | 4, 5 | mpbiri 260 | . 2 ⊢ (𝐹 = ∅ → 𝐹:∅⟶𝑋) |
7 | 3, 6 | impbii 211 | 1 ⊢ (𝐹:∅⟶𝑋 ↔ 𝐹 = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 = wceq 1537 ∅c0 4293 Fn wfn 6352 ⟶wf 6353 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pr 5332 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-br 5069 df-opab 5131 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-fun 6359 df-fn 6360 df-f 6361 |
This theorem is referenced by: f0dom0 6565 mapdm0 8423 0map0sn0 8451 griedg0ssusgr 27049 rgrusgrprc 27373 2ffzoeq 43535 |
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