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Mirrors > Home > ILE Home > Th. List > dffn2 | GIF version |
Description: Any function is a mapping into V. (Contributed by NM, 31-Oct-1995.) (Proof shortened by Andrew Salmon, 17-Sep-2011.) |
Ref | Expression |
---|---|
dffn2 | ⊢ (𝐹 Fn 𝐴 ↔ 𝐹:𝐴⟶V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssv 3169 | . . 3 ⊢ ran 𝐹 ⊆ V | |
2 | 1 | biantru 300 | . 2 ⊢ (𝐹 Fn 𝐴 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ V)) |
3 | df-f 5200 | . 2 ⊢ (𝐹:𝐴⟶V ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ V)) | |
4 | 2, 3 | bitr4i 186 | 1 ⊢ (𝐹 Fn 𝐴 ↔ 𝐹:𝐴⟶V) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 ↔ wb 104 Vcvv 2730 ⊆ wss 3121 ran crn 4610 Fn wfn 5191 ⟶wf 5192 |
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-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-11 1499 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-ext 2152 |
This theorem depends on definitions: df-bi 116 df-nf 1454 df-sb 1756 df-clab 2157 df-cleq 2163 df-clel 2166 df-v 2732 df-in 3127 df-ss 3134 df-f 5200 |
This theorem is referenced by: f1cnvcnv 5412 fcoconst 5665 fnressn 5680 1stcof 6140 2ndcof 6141 fnmpo 6179 tposfn 6250 tfrlemibfn 6305 tfr1onlembfn 6321 mptelixpg 6710 |
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