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Mirrors > Home > ILE Home > Th. List > fnssresb | GIF version |
Description: Restriction of a function with a subclass of its domain. (Contributed by NM, 10-Oct-2007.) |
Ref | Expression |
---|---|
fnssresb | ⊢ (𝐹 Fn A → ((𝐹 ↾ B) Fn B ↔ B ⊆ A)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-fn 4848 | . 2 ⊢ ((𝐹 ↾ B) Fn B ↔ (Fun (𝐹 ↾ B) ∧ dom (𝐹 ↾ B) = B)) | |
2 | fnfun 4939 | . . . . 5 ⊢ (𝐹 Fn A → Fun 𝐹) | |
3 | funres 4884 | . . . . 5 ⊢ (Fun 𝐹 → Fun (𝐹 ↾ B)) | |
4 | 2, 3 | syl 14 | . . . 4 ⊢ (𝐹 Fn A → Fun (𝐹 ↾ B)) |
5 | 4 | biantrurd 289 | . . 3 ⊢ (𝐹 Fn A → (dom (𝐹 ↾ B) = B ↔ (Fun (𝐹 ↾ B) ∧ dom (𝐹 ↾ B) = B))) |
6 | ssdmres 4576 | . . . 4 ⊢ (B ⊆ dom 𝐹 ↔ dom (𝐹 ↾ B) = B) | |
7 | fndm 4941 | . . . . 5 ⊢ (𝐹 Fn A → dom 𝐹 = A) | |
8 | 7 | sseq2d 2967 | . . . 4 ⊢ (𝐹 Fn A → (B ⊆ dom 𝐹 ↔ B ⊆ A)) |
9 | 6, 8 | syl5bbr 183 | . . 3 ⊢ (𝐹 Fn A → (dom (𝐹 ↾ B) = B ↔ B ⊆ A)) |
10 | 5, 9 | bitr3d 179 | . 2 ⊢ (𝐹 Fn A → ((Fun (𝐹 ↾ B) ∧ dom (𝐹 ↾ B) = B) ↔ B ⊆ A)) |
11 | 1, 10 | syl5bb 181 | 1 ⊢ (𝐹 Fn A → ((𝐹 ↾ B) Fn B ↔ B ⊆ A)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 ↔ wb 98 = wceq 1242 ⊆ wss 2911 dom cdm 4288 ↾ cres 4290 Fun wfun 4839 Fn wfn 4840 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-14 1402 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 ax-sep 3866 ax-pow 3918 ax-pr 3935 |
This theorem depends on definitions: df-bi 110 df-3an 886 df-tru 1245 df-nf 1347 df-sb 1643 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-ral 2305 df-rex 2306 df-v 2553 df-un 2916 df-in 2918 df-ss 2925 df-pw 3353 df-sn 3373 df-pr 3374 df-op 3376 df-br 3756 df-opab 3810 df-xp 4294 df-rel 4295 df-cnv 4296 df-co 4297 df-dm 4298 df-res 4300 df-fun 4847 df-fn 4848 |
This theorem is referenced by: fnssres 4955 |
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