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Mirrors > Home > MPE Home > Th. List > resdm | Structured version Visualization version GIF version |
Description: A relation restricted to its domain equals itself. (Contributed by NM, 12-Dec-2006.) |
Ref | Expression |
---|---|
resdm | ⊢ (Rel 𝐴 → (𝐴 ↾ dom 𝐴) = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssid 3986 | . 2 ⊢ dom 𝐴 ⊆ dom 𝐴 | |
2 | relssres 5886 | . 2 ⊢ ((Rel 𝐴 ∧ dom 𝐴 ⊆ dom 𝐴) → (𝐴 ↾ dom 𝐴) = 𝐴) | |
3 | 1, 2 | mpan2 687 | 1 ⊢ (Rel 𝐴 → (𝐴 ↾ dom 𝐴) = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1528 ⊆ wss 3933 dom cdm 5548 ↾ cres 5550 Rel wrel 5553 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-br 5058 df-opab 5120 df-xp 5554 df-rel 5555 df-dm 5558 df-res 5560 |
This theorem is referenced by: resindm 5893 resdm2 6081 relresfld 6120 fimadmfoALT 6594 fnex 6971 dftpos2 7898 tfrlem11 8013 tfrlem15 8017 tfrlem16 8018 pmresg 8423 domss2 8664 axdc3lem4 9863 gruima 10212 reldisjun 30281 funresdm1 30283 bnj1321 32196 funsseq 32908 nosupbnd2lem1 33112 nosupbnd2 33113 noetalem2 33115 noetalem3 33116 alrmomodm 35494 relbrcoss 35566 unidmqs 35768 releldmqs 35772 releldmqscoss 35774 seff 40518 sblpnf 40519 |
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