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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 102 | . . . 4 ⊢ ((Rel A ∧ dom A ⊆ B) → Rel A) | |
2 | vex 2554 | . . . . . . . . 9 ⊢ x ∈ V | |
3 | vex 2554 | . . . . . . . . 9 ⊢ y ∈ V | |
4 | 2, 3 | opeldm 4481 | . . . . . . . 8 ⊢ (〈x, y〉 ∈ A → x ∈ dom A) |
5 | ssel 2933 | . . . . . . . 8 ⊢ (dom A ⊆ B → (x ∈ dom A → x ∈ B)) | |
6 | 4, 5 | syl5 28 | . . . . . . 7 ⊢ (dom A ⊆ B → (〈x, y〉 ∈ A → x ∈ B)) |
7 | 6 | ancld 308 | . . . . . 6 ⊢ (dom A ⊆ B → (〈x, y〉 ∈ A → (〈x, y〉 ∈ A ∧ x ∈ B))) |
8 | 3 | opelres 4560 | . . . . . 6 ⊢ (〈x, y〉 ∈ (A ↾ B) ↔ (〈x, y〉 ∈ A ∧ x ∈ B)) |
9 | 7, 8 | syl6ibr 151 | . . . . 5 ⊢ (dom A ⊆ B → (〈x, y〉 ∈ A → 〈x, y〉 ∈ (A ↾ B))) |
10 | 9 | adantl 262 | . . . 4 ⊢ ((Rel A ∧ dom A ⊆ B) → (〈x, y〉 ∈ A → 〈x, y〉 ∈ (A ↾ B))) |
11 | 1, 10 | relssdv 4375 | . . 3 ⊢ ((Rel A ∧ dom A ⊆ B) → A ⊆ (A ↾ B)) |
12 | resss 4578 | . . 3 ⊢ (A ↾ B) ⊆ A | |
13 | 11, 12 | jctil 295 | . 2 ⊢ ((Rel A ∧ dom A ⊆ B) → ((A ↾ B) ⊆ A ∧ A ⊆ (A ↾ B))) |
14 | eqss 2954 | . 2 ⊢ ((A ↾ B) = A ↔ ((A ↾ B) ⊆ A ∧ A ⊆ (A ↾ B))) | |
15 | 13, 14 | sylibr 137 | 1 ⊢ ((Rel A ∧ dom A ⊆ B) → (A ↾ B) = A) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 = wceq 1242 ∈ wcel 1390 ⊆ wss 2911 〈cop 3370 dom cdm 4288 ↾ cres 4290 Rel wrel 4293 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-14 1402 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 ax-sep 3866 ax-pow 3918 ax-pr 3935 |
This theorem depends on definitions: df-bi 110 df-3an 886 df-tru 1245 df-nf 1347 df-sb 1643 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-ral 2305 df-rex 2306 df-v 2553 df-un 2916 df-in 2918 df-ss 2925 df-pw 3353 df-sn 3373 df-pr 3374 df-op 3376 df-br 3756 df-opab 3810 df-xp 4294 df-rel 4295 df-dm 4298 df-res 4300 |
This theorem is referenced by: resdm 4592 resid 4605 fnresdm 4951 f1ompt 5263 |
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