![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > fnresdm | GIF version |
Description: A function does not change when restricted to its domain. (Contributed by NM, 5-Sep-2004.) |
Ref | Expression |
---|---|
fnresdm | ⊢ (𝐹 Fn 𝐴 → (𝐹 ↾ 𝐴) = 𝐹) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fnrel 5229 | . 2 ⊢ (𝐹 Fn 𝐴 → Rel 𝐹) | |
2 | fndm 5230 | . . 3 ⊢ (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴) | |
3 | eqimss 3156 | . . 3 ⊢ (dom 𝐹 = 𝐴 → dom 𝐹 ⊆ 𝐴) | |
4 | 2, 3 | syl 14 | . 2 ⊢ (𝐹 Fn 𝐴 → dom 𝐹 ⊆ 𝐴) |
5 | relssres 4865 | . 2 ⊢ ((Rel 𝐹 ∧ dom 𝐹 ⊆ 𝐴) → (𝐹 ↾ 𝐴) = 𝐹) | |
6 | 1, 4, 5 | syl2anc 409 | 1 ⊢ (𝐹 Fn 𝐴 → (𝐹 ↾ 𝐴) = 𝐹) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1332 ⊆ wss 3076 dom cdm 4547 ↾ cres 4549 Rel wrel 4552 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-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-xp 4553 df-rel 4554 df-dm 4557 df-res 4559 df-fun 5133 df-fn 5134 |
This theorem is referenced by: fnima 5249 fresin 5309 resasplitss 5310 fnsnsplitss 5627 fsnunfv 5629 fsnunres 5630 fnsnsplitdc 6409 fnfi 6833 fseq1p1m1 9905 facnn 10505 fac0 10506 dfrelog 12989 |
Copyright terms: Public domain | W3C validator |