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Mirrors > Home > ILE Home > Th. List > relssres | GIF version |
Description: Simplification law for restriction. (Contributed by NM, 16-Aug-1994.) |
Ref | Expression |
---|---|
relssres | ⊢ ((Rel 𝐴 ∧ dom 𝐴 ⊆ 𝐵) → (𝐴 ↾ 𝐵) = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl 108 | . . . 4 ⊢ ((Rel 𝐴 ∧ dom 𝐴 ⊆ 𝐵) → Rel 𝐴) | |
2 | vex 2689 | . . . . . . . . 9 ⊢ 𝑥 ∈ V | |
3 | vex 2689 | . . . . . . . . 9 ⊢ 𝑦 ∈ V | |
4 | 2, 3 | opeldm 4742 | . . . . . . . 8 ⊢ (〈𝑥, 𝑦〉 ∈ 𝐴 → 𝑥 ∈ dom 𝐴) |
5 | ssel 3091 | . . . . . . . 8 ⊢ (dom 𝐴 ⊆ 𝐵 → (𝑥 ∈ dom 𝐴 → 𝑥 ∈ 𝐵)) | |
6 | 4, 5 | syl5 32 | . . . . . . 7 ⊢ (dom 𝐴 ⊆ 𝐵 → (〈𝑥, 𝑦〉 ∈ 𝐴 → 𝑥 ∈ 𝐵)) |
7 | 6 | ancld 323 | . . . . . 6 ⊢ (dom 𝐴 ⊆ 𝐵 → (〈𝑥, 𝑦〉 ∈ 𝐴 → (〈𝑥, 𝑦〉 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵))) |
8 | 3 | opelres 4824 | . . . . . 6 ⊢ (〈𝑥, 𝑦〉 ∈ (𝐴 ↾ 𝐵) ↔ (〈𝑥, 𝑦〉 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)) |
9 | 7, 8 | syl6ibr 161 | . . . . 5 ⊢ (dom 𝐴 ⊆ 𝐵 → (〈𝑥, 𝑦〉 ∈ 𝐴 → 〈𝑥, 𝑦〉 ∈ (𝐴 ↾ 𝐵))) |
10 | 9 | adantl 275 | . . . 4 ⊢ ((Rel 𝐴 ∧ dom 𝐴 ⊆ 𝐵) → (〈𝑥, 𝑦〉 ∈ 𝐴 → 〈𝑥, 𝑦〉 ∈ (𝐴 ↾ 𝐵))) |
11 | 1, 10 | relssdv 4631 | . . 3 ⊢ ((Rel 𝐴 ∧ dom 𝐴 ⊆ 𝐵) → 𝐴 ⊆ (𝐴 ↾ 𝐵)) |
12 | resss 4843 | . . 3 ⊢ (𝐴 ↾ 𝐵) ⊆ 𝐴 | |
13 | 11, 12 | jctil 310 | . 2 ⊢ ((Rel 𝐴 ∧ dom 𝐴 ⊆ 𝐵) → ((𝐴 ↾ 𝐵) ⊆ 𝐴 ∧ 𝐴 ⊆ (𝐴 ↾ 𝐵))) |
14 | eqss 3112 | . 2 ⊢ ((𝐴 ↾ 𝐵) = 𝐴 ↔ ((𝐴 ↾ 𝐵) ⊆ 𝐴 ∧ 𝐴 ⊆ (𝐴 ↾ 𝐵))) | |
15 | 13, 14 | sylibr 133 | 1 ⊢ ((Rel 𝐴 ∧ dom 𝐴 ⊆ 𝐵) → (𝐴 ↾ 𝐵) = 𝐴) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1331 ∈ wcel 1480 ⊆ wss 3071 〈cop 3530 dom cdm 4539 ↾ cres 4541 Rel wrel 4544 |
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 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-v 2688 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-br 3930 df-opab 3990 df-xp 4545 df-rel 4546 df-dm 4549 df-res 4551 |
This theorem is referenced by: resdm 4858 resid 4875 fnresdm 5232 f1ompt 5571 setscom 11999 setsslid 12009 |
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