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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 5201 | . 2 ⊢ ((𝐹 ↾ 𝐵) Fn 𝐵 ↔ (Fun (𝐹 ↾ 𝐵) ∧ dom (𝐹 ↾ 𝐵) = 𝐵)) | |
2 | fnfun 5295 | . . . . 5 ⊢ (𝐹 Fn 𝐴 → Fun 𝐹) | |
3 | funres 5239 | . . . . 5 ⊢ (Fun 𝐹 → Fun (𝐹 ↾ 𝐵)) | |
4 | 2, 3 | syl 14 | . . . 4 ⊢ (𝐹 Fn 𝐴 → Fun (𝐹 ↾ 𝐵)) |
5 | 4 | biantrurd 303 | . . 3 ⊢ (𝐹 Fn 𝐴 → (dom (𝐹 ↾ 𝐵) = 𝐵 ↔ (Fun (𝐹 ↾ 𝐵) ∧ dom (𝐹 ↾ 𝐵) = 𝐵))) |
6 | ssdmres 4913 | . . . 4 ⊢ (𝐵 ⊆ dom 𝐹 ↔ dom (𝐹 ↾ 𝐵) = 𝐵) | |
7 | fndm 5297 | . . . . 5 ⊢ (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴) | |
8 | 7 | sseq2d 3177 | . . . 4 ⊢ (𝐹 Fn 𝐴 → (𝐵 ⊆ dom 𝐹 ↔ 𝐵 ⊆ 𝐴)) |
9 | 6, 8 | bitr3id 193 | . . 3 ⊢ (𝐹 Fn 𝐴 → (dom (𝐹 ↾ 𝐵) = 𝐵 ↔ 𝐵 ⊆ 𝐴)) |
10 | 5, 9 | bitr3d 189 | . 2 ⊢ (𝐹 Fn 𝐴 → ((Fun (𝐹 ↾ 𝐵) ∧ dom (𝐹 ↾ 𝐵) = 𝐵) ↔ 𝐵 ⊆ 𝐴)) |
11 | 1, 10 | syl5bb 191 | 1 ⊢ (𝐹 Fn 𝐴 → ((𝐹 ↾ 𝐵) Fn 𝐵 ↔ 𝐵 ⊆ 𝐴)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 = wceq 1348 ⊆ wss 3121 dom cdm 4611 ↾ cres 4613 Fun wfun 5192 Fn wfn 5193 |
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 4107 ax-pow 4160 ax-pr 4194 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-nf 1454 df-sb 1756 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 3568 df-sn 3589 df-pr 3590 df-op 3592 df-br 3990 df-opab 4051 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-res 4623 df-fun 5200 df-fn 5201 |
This theorem is referenced by: fnssres 5311 |
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