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Mirrors > Home > ILE Home > Th. List > relres | GIF version |
Description: A restriction is a relation. Exercise 12 of [TakeutiZaring] p. 25. (Contributed by NM, 2-Aug-1994.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) |
Ref | Expression |
---|---|
relres | ⊢ Rel (𝐴 ↾ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-res 4656 | . . 3 ⊢ (𝐴 ↾ 𝐵) = (𝐴 ∩ (𝐵 × V)) | |
2 | inss2 3371 | . . 3 ⊢ (𝐴 ∩ (𝐵 × V)) ⊆ (𝐵 × V) | |
3 | 1, 2 | eqsstri 3202 | . 2 ⊢ (𝐴 ↾ 𝐵) ⊆ (𝐵 × V) |
4 | relxp 4753 | . 2 ⊢ Rel (𝐵 × V) | |
5 | relss 4731 | . 2 ⊢ ((𝐴 ↾ 𝐵) ⊆ (𝐵 × V) → (Rel (𝐵 × V) → Rel (𝐴 ↾ 𝐵))) | |
6 | 3, 4, 5 | mp2 16 | 1 ⊢ Rel (𝐴 ↾ 𝐵) |
Colors of variables: wff set class |
Syntax hints: Vcvv 2752 ∩ cin 3143 ⊆ wss 3144 × cxp 4642 ↾ cres 4646 Rel wrel 4649 |
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-ext 2171 |
This theorem depends on definitions: df-bi 117 df-tru 1367 df-nf 1472 df-sb 1774 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-v 2754 df-in 3150 df-ss 3157 df-opab 4080 df-xp 4650 df-rel 4651 df-res 4656 |
This theorem is referenced by: elres 4961 resiexg 4970 iss 4971 dfres2 4977 restidsing 4981 issref 5029 asymref 5032 poirr2 5039 cnvcnvres 5110 resco 5151 ressn 5187 funssres 5277 fnresdisj 5345 fnres 5351 fcnvres 5418 nfunsn 5568 fsnunfv 5737 resfunexgALT 6132 setsresg 12549 |
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