Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > fnresi | Structured version Visualization version GIF version |
Description: Functionality and domain of restricted identity. (Contributed by NM, 27-Aug-2004.) |
Ref | Expression |
---|---|
fnresi | ⊢ ( I ↾ 𝐴) Fn 𝐴 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | funi 5834 | . . 3 ⊢ Fun I | |
2 | funres 5843 | . . 3 ⊢ (Fun I → Fun ( I ↾ 𝐴)) | |
3 | 1, 2 | ax-mp 5 | . 2 ⊢ Fun ( I ↾ 𝐴) |
4 | dmresi 5376 | . 2 ⊢ dom ( I ↾ 𝐴) = 𝐴 | |
5 | df-fn 5807 | . 2 ⊢ (( I ↾ 𝐴) Fn 𝐴 ↔ (Fun ( I ↾ 𝐴) ∧ dom ( I ↾ 𝐴) = 𝐴)) | |
6 | 3, 4, 5 | mpbir2an 957 | 1 ⊢ ( I ↾ 𝐴) Fn 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1475 I cid 4948 dom cdm 5038 ↾ cres 5040 Fun wfun 5798 Fn wfn 5799 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-br 4584 df-opab 4644 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-res 5050 df-fun 5806 df-fn 5807 |
This theorem is referenced by: f1oi 6086 fninfp 6345 fndifnfp 6347 fnnfpeq0 6349 fveqf1o 6457 weniso 6504 iordsmo 7341 fipreima 8155 dfac9 8841 pmtrfinv 17704 ustuqtop3 21857 fta1blem 23732 qaa 23882 dfiop2 27996 idssxp 28811 cvmliftlem4 30524 cvmliftlem5 30525 poimirlem15 32594 poimirlem22 32601 ltrnid 34439 rtrclex 36943 dvsid 37552 dflinc2 41993 |
Copyright terms: Public domain | W3C validator |