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Mirrors > Home > ILE Home > Th. List > fresin | GIF version |
Description: An identity for the mapping relationship under restriction. (Contributed by Scott Fenton, 4-Sep-2011.) (Proof shortened by Mario Carneiro, 26-May-2016.) |
Ref | Expression |
---|---|
fresin | ⊢ (𝐹:𝐴⟶𝐵 → (𝐹 ↾ 𝑋):(𝐴 ∩ 𝑋)⟶𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | inss1 3379 | . . 3 ⊢ (𝐴 ∩ 𝑋) ⊆ 𝐴 | |
2 | fssres 5429 | . . 3 ⊢ ((𝐹:𝐴⟶𝐵 ∧ (𝐴 ∩ 𝑋) ⊆ 𝐴) → (𝐹 ↾ (𝐴 ∩ 𝑋)):(𝐴 ∩ 𝑋)⟶𝐵) | |
3 | 1, 2 | mpan2 425 | . 2 ⊢ (𝐹:𝐴⟶𝐵 → (𝐹 ↾ (𝐴 ∩ 𝑋)):(𝐴 ∩ 𝑋)⟶𝐵) |
4 | resres 4954 | . . . 4 ⊢ ((𝐹 ↾ 𝐴) ↾ 𝑋) = (𝐹 ↾ (𝐴 ∩ 𝑋)) | |
5 | ffn 5403 | . . . . . 6 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹 Fn 𝐴) | |
6 | fnresdm 5363 | . . . . . 6 ⊢ (𝐹 Fn 𝐴 → (𝐹 ↾ 𝐴) = 𝐹) | |
7 | 5, 6 | syl 14 | . . . . 5 ⊢ (𝐹:𝐴⟶𝐵 → (𝐹 ↾ 𝐴) = 𝐹) |
8 | 7 | reseq1d 4941 | . . . 4 ⊢ (𝐹:𝐴⟶𝐵 → ((𝐹 ↾ 𝐴) ↾ 𝑋) = (𝐹 ↾ 𝑋)) |
9 | 4, 8 | eqtr3id 2240 | . . 3 ⊢ (𝐹:𝐴⟶𝐵 → (𝐹 ↾ (𝐴 ∩ 𝑋)) = (𝐹 ↾ 𝑋)) |
10 | 9 | feq1d 5390 | . 2 ⊢ (𝐹:𝐴⟶𝐵 → ((𝐹 ↾ (𝐴 ∩ 𝑋)):(𝐴 ∩ 𝑋)⟶𝐵 ↔ (𝐹 ↾ 𝑋):(𝐴 ∩ 𝑋)⟶𝐵)) |
11 | 3, 10 | mpbid 147 | 1 ⊢ (𝐹:𝐴⟶𝐵 → (𝐹 ↾ 𝑋):(𝐴 ∩ 𝑋)⟶𝐵) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1364 ∩ cin 3152 ⊆ wss 3153 ↾ cres 4661 Fn wfn 5249 ⟶wf 5250 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-14 2167 ax-ext 2175 ax-sep 4147 ax-pow 4203 ax-pr 4238 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-nf 1472 df-sb 1774 df-clab 2180 df-cleq 2186 df-clel 2189 df-nfc 2325 df-ral 2477 df-rex 2478 df-v 2762 df-un 3157 df-in 3159 df-ss 3166 df-pw 3603 df-sn 3624 df-pr 3625 df-op 3627 df-br 4030 df-opab 4091 df-xp 4665 df-rel 4666 df-cnv 4667 df-co 4668 df-dm 4669 df-rn 4670 df-res 4671 df-fun 5256 df-fn 5257 df-f 5258 |
This theorem is referenced by: limcresi 14820 |
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