![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 4018 | . 2 ⊢ dom 𝐴 ⊆ dom 𝐴 | |
2 | relssres 6042 | . 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 1537 ⊆ wss 3963 dom cdm 5689 ↾ cres 5691 Rel wrel 5694 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-br 5149 df-opab 5211 df-xp 5695 df-rel 5696 df-dm 5699 df-res 5701 |
This theorem is referenced by: resindm 6050 reldisjun 6052 relresdm1 6053 resdm2 6253 relresfld 6298 fimadmfoALT 6832 fnex 7237 dftpos2 8267 tfrlem11 8427 tfrlem15 8431 tfrlem16 8432 pmresg 8909 domss2 9175 axdc3lem4 10491 gruima 10840 nosupbnd2lem1 27775 nosupbnd2 27776 noinfbnd2lem1 27790 noinfbnd2 27791 noetasuplem2 27794 noetasuplem3 27795 noetasuplem4 27796 noetainflem2 27798 bnj1321 35020 funsseq 35749 alrmomodm 38341 relbrcoss 38428 unidmqs 38636 releldmqs 38640 releldmqscoss 38642 seff 44305 sblpnf 44306 f1cof1blem 47024 funfocofob 47028 itcoval1 48513 |
Copyright terms: Public domain | W3C validator |