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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 3337 | . . 3 ⊢ (𝐴 ∩ 𝑋) ⊆ 𝐴 | |
2 | fssres 5357 | . . 3 ⊢ ((𝐹:𝐴⟶𝐵 ∧ (𝐴 ∩ 𝑋) ⊆ 𝐴) → (𝐹 ↾ (𝐴 ∩ 𝑋)):(𝐴 ∩ 𝑋)⟶𝐵) | |
3 | 1, 2 | mpan2 422 | . 2 ⊢ (𝐹:𝐴⟶𝐵 → (𝐹 ↾ (𝐴 ∩ 𝑋)):(𝐴 ∩ 𝑋)⟶𝐵) |
4 | resres 4890 | . . . 4 ⊢ ((𝐹 ↾ 𝐴) ↾ 𝑋) = (𝐹 ↾ (𝐴 ∩ 𝑋)) | |
5 | ffn 5331 | . . . . . 6 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹 Fn 𝐴) | |
6 | fnresdm 5291 | . . . . . 6 ⊢ (𝐹 Fn 𝐴 → (𝐹 ↾ 𝐴) = 𝐹) | |
7 | 5, 6 | syl 14 | . . . . 5 ⊢ (𝐹:𝐴⟶𝐵 → (𝐹 ↾ 𝐴) = 𝐹) |
8 | 7 | reseq1d 4877 | . . . 4 ⊢ (𝐹:𝐴⟶𝐵 → ((𝐹 ↾ 𝐴) ↾ 𝑋) = (𝐹 ↾ 𝑋)) |
9 | 4, 8 | eqtr3id 2211 | . . 3 ⊢ (𝐹:𝐴⟶𝐵 → (𝐹 ↾ (𝐴 ∩ 𝑋)) = (𝐹 ↾ 𝑋)) |
10 | 9 | feq1d 5318 | . 2 ⊢ (𝐹:𝐴⟶𝐵 → ((𝐹 ↾ (𝐴 ∩ 𝑋)):(𝐴 ∩ 𝑋)⟶𝐵 ↔ (𝐹 ↾ 𝑋):(𝐴 ∩ 𝑋)⟶𝐵)) |
11 | 3, 10 | mpbid 146 | 1 ⊢ (𝐹:𝐴⟶𝐵 → (𝐹 ↾ 𝑋):(𝐴 ∩ 𝑋)⟶𝐵) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1342 ∩ cin 3110 ⊆ wss 3111 ↾ cres 4600 Fn wfn 5177 ⟶wf 5178 |
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 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-pow 4147 ax-pr 4181 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-nf 1448 df-sb 1750 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ral 2447 df-rex 2448 df-v 2723 df-un 3115 df-in 3117 df-ss 3124 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-br 3977 df-opab 4038 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-rn 4609 df-res 4610 df-fun 5184 df-fn 5185 df-f 5186 |
This theorem is referenced by: limcresi 13176 |
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