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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 2692 | . . . . . . . . 9 ⊢ 𝑥 ∈ V | |
3 | vex 2692 | . . . . . . . . 9 ⊢ 𝑦 ∈ V | |
4 | 2, 3 | opeldm 4750 | . . . . . . . 8 ⊢ (〈𝑥, 𝑦〉 ∈ 𝐴 → 𝑥 ∈ dom 𝐴) |
5 | ssel 3096 | . . . . . . . 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 4832 | . . . . . 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 4639 | . . 3 ⊢ ((Rel 𝐴 ∧ dom 𝐴 ⊆ 𝐵) → 𝐴 ⊆ (𝐴 ↾ 𝐵)) |
12 | resss 4851 | . . 3 ⊢ (𝐴 ↾ 𝐵) ⊆ 𝐴 | |
13 | 11, 12 | jctil 310 | . 2 ⊢ ((Rel 𝐴 ∧ dom 𝐴 ⊆ 𝐵) → ((𝐴 ↾ 𝐵) ⊆ 𝐴 ∧ 𝐴 ⊆ (𝐴 ↾ 𝐵))) |
14 | eqss 3117 | . 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 1332 ∈ wcel 1481 ⊆ wss 3076 〈cop 3535 dom cdm 4547 ↾ cres 4549 Rel wrel 4552 |
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 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-pow 4106 ax-pr 4139 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-nf 1438 df-sb 1737 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ral 2422 df-rex 2423 df-v 2691 df-un 3080 df-in 3082 df-ss 3089 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-br 3938 df-opab 3998 df-xp 4553 df-rel 4554 df-dm 4557 df-res 4559 |
This theorem is referenced by: resdm 4866 resid 4883 fnresdm 5240 f1ompt 5579 setscom 12038 setsslid 12048 |
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