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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 4357 | . . 3 ⊢ (𝐴 ↾ 𝐵) = (𝐴 ∩ (𝐵 × V)) | |
2 | inss2 3158 | . . 3 ⊢ (𝐴 ∩ (𝐵 × V)) ⊆ (𝐵 × V) | |
3 | 1, 2 | eqsstri 2975 | . 2 ⊢ (𝐴 ↾ 𝐵) ⊆ (𝐵 × V) |
4 | relxp 4447 | . 2 ⊢ Rel (𝐵 × V) | |
5 | relss 4427 | . 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 2557 ∩ cin 2916 ⊆ wss 2917 × cxp 4343 ↾ cres 4347 Rel wrel 4350 |
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 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 |
This theorem depends on definitions: df-bi 110 df-tru 1246 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-v 2559 df-in 2924 df-ss 2931 df-opab 3819 df-xp 4351 df-rel 4352 df-res 4357 |
This theorem is referenced by: elres 4646 resiexg 4653 iss 4654 dfres2 4658 issref 4707 asymref 4710 poirr2 4717 cnvcnvres 4784 resco 4825 ressn 4858 funssres 4942 fnresdisj 5009 fnres 5015 fcnvres 5073 nfunsn 5207 fsnunfv 5363 resfunexgALT 5737 |
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