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Mirrors > Home > ILE Home > Th. List > f0bi | 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 5360 | . . 3 ⊢ (𝐹:∅⟶𝑋 → 𝐹 Fn ∅) | |
2 | fn0 5330 | . . 3 ⊢ (𝐹 Fn ∅ ↔ 𝐹 = ∅) | |
3 | 1, 2 | sylib 122 | . 2 ⊢ (𝐹:∅⟶𝑋 → 𝐹 = ∅) |
4 | f0 5401 | . . 3 ⊢ ∅:∅⟶𝑋 | |
5 | feq1 5343 | . . 3 ⊢ (𝐹 = ∅ → (𝐹:∅⟶𝑋 ↔ ∅:∅⟶𝑋)) | |
6 | 4, 5 | mpbiri 168 | . 2 ⊢ (𝐹 = ∅ → 𝐹:∅⟶𝑋) |
7 | 3, 6 | impbii 126 | 1 ⊢ (𝐹:∅⟶𝑋 ↔ 𝐹 = ∅) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 105 = wceq 1353 ∅c0 3422 Fn wfn 5206 ⟶wf 5207 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-14 2151 ax-ext 2159 ax-sep 4118 ax-nul 4126 ax-pow 4171 ax-pr 4205 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-rex 2461 df-v 2739 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-nul 3423 df-pw 3576 df-sn 3597 df-pr 3598 df-op 3600 df-br 4001 df-opab 4062 df-id 4289 df-xp 4628 df-rel 4629 df-cnv 4630 df-co 4631 df-dm 4632 df-rn 4633 df-fun 5213 df-fn 5214 df-f 5215 |
This theorem is referenced by: f0dom0 5404 mapdm0 6656 map0e 6679 |
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