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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 5272 | . . 3 ⊢ (𝐹:∅⟶𝑋 → 𝐹 Fn ∅) | |
2 | fn0 5242 | . . 3 ⊢ (𝐹 Fn ∅ ↔ 𝐹 = ∅) | |
3 | 1, 2 | sylib 121 | . 2 ⊢ (𝐹:∅⟶𝑋 → 𝐹 = ∅) |
4 | f0 5313 | . . 3 ⊢ ∅:∅⟶𝑋 | |
5 | feq1 5255 | . . 3 ⊢ (𝐹 = ∅ → (𝐹:∅⟶𝑋 ↔ ∅:∅⟶𝑋)) | |
6 | 4, 5 | mpbiri 167 | . 2 ⊢ (𝐹 = ∅ → 𝐹:∅⟶𝑋) |
7 | 3, 6 | impbii 125 | 1 ⊢ (𝐹:∅⟶𝑋 ↔ 𝐹 = ∅) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 104 = wceq 1331 ∅c0 3363 Fn wfn 5118 ⟶wf 5119 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-nul 4054 ax-pow 4098 ax-pr 4131 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-v 2688 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-br 3930 df-opab 3990 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-fun 5125 df-fn 5126 df-f 5127 |
This theorem is referenced by: f0dom0 5316 mapdm0 6557 map0e 6580 |
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