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Mirrors > Home > ILE Home > Th. List > fssres2 | GIF version |
Description: Restriction of a restricted function with a subclass of its domain. (Contributed by NM, 21-Jul-2005.) |
Ref | Expression |
---|---|
fssres2 | ⊢ (((𝐹 ↾ 𝐴):𝐴⟶𝐵 ∧ 𝐶 ⊆ 𝐴) → (𝐹 ↾ 𝐶):𝐶⟶𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fssres 5268 | . 2 ⊢ (((𝐹 ↾ 𝐴):𝐴⟶𝐵 ∧ 𝐶 ⊆ 𝐴) → ((𝐹 ↾ 𝐴) ↾ 𝐶):𝐶⟶𝐵) | |
2 | resabs1 4818 | . . . 4 ⊢ (𝐶 ⊆ 𝐴 → ((𝐹 ↾ 𝐴) ↾ 𝐶) = (𝐹 ↾ 𝐶)) | |
3 | 2 | feq1d 5229 | . . 3 ⊢ (𝐶 ⊆ 𝐴 → (((𝐹 ↾ 𝐴) ↾ 𝐶):𝐶⟶𝐵 ↔ (𝐹 ↾ 𝐶):𝐶⟶𝐵)) |
4 | 3 | adantl 275 | . 2 ⊢ (((𝐹 ↾ 𝐴):𝐴⟶𝐵 ∧ 𝐶 ⊆ 𝐴) → (((𝐹 ↾ 𝐴) ↾ 𝐶):𝐶⟶𝐵 ↔ (𝐹 ↾ 𝐶):𝐶⟶𝐵)) |
5 | 1, 4 | mpbid 146 | 1 ⊢ (((𝐹 ↾ 𝐴):𝐴⟶𝐵 ∧ 𝐶 ⊆ 𝐴) → (𝐹 ↾ 𝐶):𝐶⟶𝐵) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 ⊆ wss 3041 ↾ cres 4511 ⟶wf 5089 |
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 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-pow 4068 ax-pr 4101 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-nf 1422 df-sb 1721 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ral 2398 df-rex 2399 df-v 2662 df-un 3045 df-in 3047 df-ss 3054 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-br 3900 df-opab 3960 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-rn 4520 df-res 4521 df-fun 5095 df-fn 5096 df-f 5097 |
This theorem is referenced by: frecsuclem 6271 |
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