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Mirrors > Home > ILE Home > Th. List > fnresi | 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 4952 | . . 3 ⊢ Fun I | |
2 | funres 4961 | . . 3 ⊢ (Fun I → Fun ( I ↾ 𝐴)) | |
3 | 1, 2 | ax-mp 7 | . 2 ⊢ Fun ( I ↾ 𝐴) |
4 | dmresi 4681 | . 2 ⊢ dom ( I ↾ 𝐴) = 𝐴 | |
5 | df-fn 4925 | . 2 ⊢ (( I ↾ 𝐴) Fn 𝐴 ↔ (Fun ( I ↾ 𝐴) ∧ dom ( I ↾ 𝐴) = 𝐴)) | |
6 | 3, 4, 5 | mpbir2an 883 | 1 ⊢ ( I ↾ 𝐴) Fn 𝐴 |
Colors of variables: wff set class |
Syntax hints: = wceq 1284 I cid 4043 dom cdm 4363 ↾ cres 4365 Fun wfun 4916 Fn wfn 4917 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-pow 3948 ax-pr 3964 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ral 2353 df-rex 2354 df-v 2603 df-un 2977 df-in 2979 df-ss 2986 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-br 3786 df-opab 3840 df-id 4048 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-res 4375 df-fun 4924 df-fn 4925 |
This theorem is referenced by: f1oi 5184 iordsmo 5935 |
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