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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 3587 | . 2 ⊢ dom 𝐴 ⊆ dom 𝐴 | |
2 | relssres 5357 | . 2 ⊢ ((Rel 𝐴 ∧ dom 𝐴 ⊆ dom 𝐴) → (𝐴 ↾ dom 𝐴) = 𝐴) | |
3 | 1, 2 | mpan2 703 | 1 ⊢ (Rel 𝐴 → (𝐴 ↾ dom 𝐴) = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ⊆ wss 3540 dom cdm 5038 ↾ cres 5040 Rel wrel 5043 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-br 4584 df-opab 4644 df-xp 5044 df-rel 5045 df-dm 5048 df-res 5050 |
This theorem is referenced by: resindm 5364 resdm2 5542 relresfld 5579 fnex 6386 dftpos2 7256 tfrlem11 7371 tfrlem15 7375 tfrlem16 7376 pmresg 7771 domss2 8004 axdc3lem4 9158 gruima 9503 bnj1321 30349 funsseq 30912 seff 37530 sblpnf 37531 resisresindm 40320 |
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