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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 5163 | . . 3 ⊢ Fun I | |
2 | funres 5172 | . . 3 ⊢ (Fun I → Fun ( I ↾ 𝐴)) | |
3 | 1, 2 | ax-mp 5 | . 2 ⊢ Fun ( I ↾ 𝐴) |
4 | dmresi 4882 | . 2 ⊢ dom ( I ↾ 𝐴) = 𝐴 | |
5 | df-fn 5134 | . 2 ⊢ (( I ↾ 𝐴) Fn 𝐴 ↔ (Fun ( I ↾ 𝐴) ∧ dom ( I ↾ 𝐴) = 𝐴)) | |
6 | 3, 4, 5 | mpbir2an 927 | 1 ⊢ ( I ↾ 𝐴) Fn 𝐴 |
Colors of variables: wff set class |
Syntax hints: = wceq 1332 I cid 4218 dom cdm 4547 ↾ cres 4549 Fun wfun 5125 Fn wfn 5126 |
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-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-pow 4106 ax-pr 4139 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ral 2422 df-rex 2423 df-v 2691 df-un 3080 df-in 3082 df-ss 3089 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-br 3938 df-opab 3998 df-id 4223 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-res 4559 df-fun 5133 df-fn 5134 |
This theorem is referenced by: f1oi 5413 iordsmo 6202 omp1eomlem 6987 ctm 7002 |
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