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Mirrors > Home > ILE Home > Th. List > resdm2 | GIF version |
Description: A class restricted to its domain equals its double converse. (Contributed by NM, 8-Apr-2007.) |
Ref | Expression |
---|---|
resdm2 | ⊢ (𝐴 ↾ dom 𝐴) = ◡◡𝐴 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rescnvcnv 4893 | . 2 ⊢ (◡◡𝐴 ↾ dom ◡◡𝐴) = (𝐴 ↾ dom ◡◡𝐴) | |
2 | relcnv 4810 | . . 3 ⊢ Rel ◡◡𝐴 | |
3 | resdm 4751 | . . 3 ⊢ (Rel ◡◡𝐴 → (◡◡𝐴 ↾ dom ◡◡𝐴) = ◡◡𝐴) | |
4 | 2, 3 | ax-mp 7 | . 2 ⊢ (◡◡𝐴 ↾ dom ◡◡𝐴) = ◡◡𝐴 |
5 | dmcnvcnv 4659 | . . 3 ⊢ dom ◡◡𝐴 = dom 𝐴 | |
6 | 5 | reseq2i 4710 | . 2 ⊢ (𝐴 ↾ dom ◡◡𝐴) = (𝐴 ↾ dom 𝐴) |
7 | 1, 4, 6 | 3eqtr3ri 2117 | 1 ⊢ (𝐴 ↾ dom 𝐴) = ◡◡𝐴 |
Colors of variables: wff set class |
Syntax hints: = wceq 1289 ◡ccnv 4437 dom cdm 4438 ↾ cres 4440 Rel wrel 4443 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 665 ax-5 1381 ax-7 1382 ax-gen 1383 ax-ie1 1427 ax-ie2 1428 ax-8 1440 ax-10 1441 ax-11 1442 ax-i12 1443 ax-bndl 1444 ax-4 1445 ax-14 1450 ax-17 1464 ax-i9 1468 ax-ial 1472 ax-i5r 1473 ax-ext 2070 ax-sep 3957 ax-pow 4009 ax-pr 4036 |
This theorem depends on definitions: df-bi 115 df-3an 926 df-tru 1292 df-nf 1395 df-sb 1693 df-eu 1951 df-mo 1952 df-clab 2075 df-cleq 2081 df-clel 2084 df-nfc 2217 df-ral 2364 df-rex 2365 df-v 2621 df-un 3003 df-in 3005 df-ss 3012 df-pw 3431 df-sn 3452 df-pr 3453 df-op 3455 df-br 3846 df-opab 3900 df-xp 4444 df-rel 4445 df-cnv 4446 df-dm 4448 df-rn 4449 df-res 4450 |
This theorem is referenced by: resdmres 4922 |
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