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| Mirrors > Home > MPE Home > Th. List > dffn2 | Structured version Visualization version 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 4008 | . . 3 ⊢ ran 𝐹 ⊆ V | |
| 2 | 1 | biantru 529 | . 2 ⊢ (𝐹 Fn 𝐴 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ V)) |
| 3 | df-f 6565 | . 2 ⊢ (𝐹:𝐴⟶V ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ V)) | |
| 4 | 2, 3 | bitr4i 278 | 1 ⊢ (𝐹 Fn 𝐴 ↔ 𝐹:𝐴⟶V) |
| Colors of variables: wff setvar class |
| Syntax hints: ↔ wb 206 ∧ wa 395 Vcvv 3480 ⊆ wss 3951 ran crn 5686 Fn wfn 6556 ⟶wf 6557 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1543 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-v 3482 df-ss 3968 df-f 6565 |
| This theorem is referenced by: f1cnvcnv 6813 fcoconst 7154 fnressn 7178 fndifnfp 7196 1stcof 8044 2ndcof 8045 fnmpo 8094 tposfn 8280 tz7.48lem 8481 seqomlem2 8491 mptelixpg 8975 r111 9815 smobeth 10626 inar1 10815 imasvscafn 17582 fucidcl 18013 fucsect 18020 dfinito3 18050 dftermo3 18051 curfcl 18277 curf2ndf 18292 dsmmbas2 21757 frlmsslsp 21816 frlmup1 21818 prdstopn 23636 prdstps 23637 ist0-4 23737 ptuncnv 23815 xpstopnlem2 23819 prdstgpd 24133 prdsxmslem2 24542 curry2ima 32718 fnchoice 45034 fsneqrn 45216 stoweidlem35 46050 fucorid2 49058 precofval2 49064 |
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