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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 4006 | . 2 ⊢ dom 𝐴 ⊆ dom 𝐴 | |
| 2 | relssres 6040 | . 2 ⊢ ((Rel 𝐴 ∧ dom 𝐴 ⊆ dom 𝐴) → (𝐴 ↾ dom 𝐴) = 𝐴) | |
| 3 | 1, 2 | mpan2 691 | 1 ⊢ (Rel 𝐴 → (𝐴 ↾ dom 𝐴) = 𝐴) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1540 ⊆ wss 3951 dom cdm 5685 ↾ cres 5687 Rel wrel 5690 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-opab 5206 df-xp 5691 df-rel 5692 df-dm 5695 df-res 5697 |
| This theorem is referenced by: resindm 6048 reldisjun 6050 relresdm1 6051 imadifssran 6171 resdm2 6251 relresfld 6296 fimadmfoALT 6831 fnex 7237 dftpos2 8268 tfrlem11 8428 tfrlem15 8432 tfrlem16 8433 pmresg 8910 domss2 9176 axdc3lem4 10493 gruima 10842 nosupbnd2lem1 27760 nosupbnd2 27761 noinfbnd2lem1 27775 noinfbnd2 27776 noetasuplem2 27779 noetasuplem3 27780 noetasuplem4 27781 noetainflem2 27783 bnj1321 35041 funsseq 35768 alrmomodm 38360 relbrcoss 38447 unidmqs 38655 releldmqs 38659 releldmqscoss 38661 seff 44328 sblpnf 44329 f1cof1blem 47086 funfocofob 47090 itcoval1 48584 |
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