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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 3192 | . . 3 ⊢ ran 𝐹 ⊆ V | |
2 | 1 | biantru 302 | . 2 ⊢ (𝐹 Fn 𝐴 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ V)) |
3 | df-f 5239 | . 2 ⊢ (𝐹:𝐴⟶V ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ V)) | |
4 | 2, 3 | bitr4i 187 | 1 ⊢ (𝐹 Fn 𝐴 ↔ 𝐹:𝐴⟶V) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 104 ↔ wb 105 Vcvv 2752 ⊆ wss 3144 ran crn 4645 Fn wfn 5230 ⟶wf 5231 |
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-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-11 1517 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-ext 2171 |
This theorem depends on definitions: df-bi 117 df-nf 1472 df-sb 1774 df-clab 2176 df-cleq 2182 df-clel 2185 df-v 2754 df-in 3150 df-ss 3157 df-f 5239 |
This theorem is referenced by: f1cnvcnv 5451 fcoconst 5708 fnressn 5723 1stcof 6188 2ndcof 6189 fnmpo 6227 tposfn 6298 tfrlemibfn 6353 tfr1onlembfn 6369 mptelixpg 6760 |
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