Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > relres | Structured version Visualization version 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 5126 | . . 3 ⊢ (𝐴 ↾ 𝐵) = (𝐴 ∩ (𝐵 × V)) | |
2 | inss2 3834 | . . 3 ⊢ (𝐴 ∩ (𝐵 × V)) ⊆ (𝐵 × V) | |
3 | 1, 2 | eqsstri 3635 | . 2 ⊢ (𝐴 ↾ 𝐵) ⊆ (𝐵 × V) |
4 | relxp 5227 | . 2 ⊢ Rel (𝐵 × V) | |
5 | relss 5206 | . 2 ⊢ ((𝐴 ↾ 𝐵) ⊆ (𝐵 × V) → (Rel (𝐵 × V) → Rel (𝐴 ↾ 𝐵))) | |
6 | 3, 4, 5 | mp2 9 | 1 ⊢ Rel (𝐴 ↾ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: Vcvv 3200 ∩ cin 3573 ⊆ wss 3574 × cxp 5112 ↾ cres 5116 Rel wrel 5119 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-v 3202 df-in 3581 df-ss 3588 df-opab 4713 df-xp 5120 df-rel 5121 df-res 5126 |
This theorem is referenced by: elres 5435 iss 5447 dfres2 5453 restidsing 5458 restidsingOLD 5459 issref 5509 asymref 5512 poirr2 5520 cnvcnvres 5598 resco 5639 coeq0 5644 ressn 5671 funssres 5930 fnresdisj 6001 fnres 6007 fresaunres2 6076 fcnvres 6082 nfunsn 6225 dffv2 6271 fsnunfv 6453 resiexg 7102 resfunexgALT 7129 domss2 8119 fidomdm 8243 dmct 9346 relexp0rel 13777 setsres 15901 pospo 16973 metustid 22359 ovoliunlem1 23270 dvres 23675 dvres2 23676 dvlog 24397 efopnlem2 24403 h2hlm 27837 hlimcaui 28093 funresdm1 29416 dfpo2 31645 eqfunresadj 31659 dfrdg2 31701 funpartfun 32050 brres2 34035 elecres 34042 mapfzcons1 37280 diophrw 37322 eldioph2lem1 37323 eldioph2lem2 37324 undmrnresiss 37910 rtrclexi 37928 brfvrcld2 37984 relexpiidm 37996 rp-imass 38065 idhe 38081 limsupresuz 39935 liminfresuz 40016 funressnfv 41208 dfdfat2 41211 setrec2lem2 42441 |
Copyright terms: Public domain | W3C validator |