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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 5228 | . . 3 ⊢ Fun I | |
2 | funres 5237 | . . 3 ⊢ (Fun I → Fun ( I ↾ 𝐴)) | |
3 | 1, 2 | ax-mp 5 | . 2 ⊢ Fun ( I ↾ 𝐴) |
4 | dmresi 4944 | . 2 ⊢ dom ( I ↾ 𝐴) = 𝐴 | |
5 | df-fn 5199 | . 2 ⊢ (( I ↾ 𝐴) Fn 𝐴 ↔ (Fun ( I ↾ 𝐴) ∧ dom ( I ↾ 𝐴) = 𝐴)) | |
6 | 3, 4, 5 | mpbir2an 937 | 1 ⊢ ( I ↾ 𝐴) Fn 𝐴 |
Colors of variables: wff set class |
Syntax hints: = wceq 1348 I cid 4271 dom cdm 4609 ↾ cres 4611 Fun wfun 5190 Fn wfn 5191 |
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 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-14 2144 ax-ext 2152 ax-sep 4105 ax-pow 4158 ax-pr 4192 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ral 2453 df-rex 2454 df-v 2732 df-un 3125 df-in 3127 df-ss 3134 df-pw 3566 df-sn 3587 df-pr 3588 df-op 3590 df-br 3988 df-opab 4049 df-id 4276 df-xp 4615 df-rel 4616 df-cnv 4617 df-co 4618 df-dm 4619 df-res 4621 df-fun 5198 df-fn 5199 |
This theorem is referenced by: f1oi 5478 iordsmo 6273 omp1eomlem 7067 ctm 7082 |
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