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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 4489 | . . 3 ⊢ (𝐴 ↾ 𝐵) = (𝐴 ∩ (𝐵 × V)) | |
2 | inss2 3244 | . . 3 ⊢ (𝐴 ∩ (𝐵 × V)) ⊆ (𝐵 × V) | |
3 | 1, 2 | eqsstri 3079 | . 2 ⊢ (𝐴 ↾ 𝐵) ⊆ (𝐵 × V) |
4 | relxp 4586 | . 2 ⊢ Rel (𝐵 × V) | |
5 | relss 4564 | . 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 2641 ∩ cin 3020 ⊆ wss 3021 × cxp 4475 ↾ cres 4479 Rel wrel 4482 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 671 ax-5 1391 ax-7 1392 ax-gen 1393 ax-ie1 1437 ax-ie2 1438 ax-8 1450 ax-10 1451 ax-11 1452 ax-i12 1453 ax-bndl 1454 ax-4 1455 ax-17 1474 ax-i9 1478 ax-ial 1482 ax-i5r 1483 ax-ext 2082 |
This theorem depends on definitions: df-bi 116 df-tru 1302 df-nf 1405 df-sb 1704 df-clab 2087 df-cleq 2093 df-clel 2096 df-nfc 2229 df-v 2643 df-in 3027 df-ss 3034 df-opab 3930 df-xp 4483 df-rel 4484 df-res 4489 |
This theorem is referenced by: elres 4791 resiexg 4800 iss 4801 dfres2 4807 issref 4857 asymref 4860 poirr2 4867 cnvcnvres 4938 resco 4979 ressn 5015 funssres 5101 fnresdisj 5169 fnres 5175 fcnvres 5242 nfunsn 5387 fsnunfv 5553 resfunexgALT 5939 setsresg 11779 |
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