![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
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 3124 | . . 3 ⊢ ran 𝐹 ⊆ V | |
2 | 1 | biantru 300 | . 2 ⊢ (𝐹 Fn 𝐴 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ V)) |
3 | df-f 5135 | . 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 2689 ⊆ wss 3076 ran crn 4548 Fn wfn 5126 ⟶wf 5127 |
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 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-11 1485 ax-4 1488 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 |
This theorem depends on definitions: df-bi 116 df-nf 1438 df-sb 1737 df-clab 2127 df-cleq 2133 df-clel 2136 df-v 2691 df-in 3082 df-ss 3089 df-f 5135 |
This theorem is referenced by: f1cnvcnv 5347 fcoconst 5599 fnressn 5614 1stcof 6069 2ndcof 6070 fnmpo 6108 tposfn 6178 tfrlemibfn 6233 tfr1onlembfn 6249 mptelixpg 6636 |
Copyright terms: Public domain | W3C validator |