![]() |
Mathbox for Peter Mazsa |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > dmcoss3 | Structured version Visualization version GIF version |
Description: The domain of cosets is the domain of converse. (Contributed by Peter Mazsa, 4-Jan-2019.) |
Ref | Expression |
---|---|
dmcoss3 | ⊢ dom ≀ 𝑅 = dom ◡𝑅 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfcoss3 38319 | . . 3 ⊢ ≀ 𝑅 = (𝑅 ∘ ◡𝑅) | |
2 | 1 | dmeqi 5928 | . 2 ⊢ dom ≀ 𝑅 = dom (𝑅 ∘ ◡𝑅) |
3 | rncnv 38205 | . . . 4 ⊢ ran ◡𝑅 = dom 𝑅 | |
4 | 3 | eqimssi 4063 | . . 3 ⊢ ran ◡𝑅 ⊆ dom 𝑅 |
5 | dmcosseq 5998 | . . 3 ⊢ (ran ◡𝑅 ⊆ dom 𝑅 → dom (𝑅 ∘ ◡𝑅) = dom ◡𝑅) | |
6 | 4, 5 | ax-mp 5 | . 2 ⊢ dom (𝑅 ∘ ◡𝑅) = dom ◡𝑅 |
7 | 2, 6 | eqtri 2762 | 1 ⊢ dom ≀ 𝑅 = dom ◡𝑅 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ⊆ wss 3970 ◡ccnv 5698 dom cdm 5699 ran crn 5700 ∘ ccom 5703 ≀ ccoss 38084 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2105 ax-9 2113 ax-10 2136 ax-12 2173 ax-ext 2705 ax-sep 5320 ax-nul 5327 ax-pr 5450 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-clab 2712 df-cleq 2726 df-clel 2813 df-rab 3439 df-v 3484 df-dif 3973 df-un 3975 df-ss 3987 df-nul 4348 df-if 4549 df-sn 4649 df-pr 4651 df-op 4655 df-br 5170 df-opab 5232 df-cnv 5707 df-co 5708 df-dm 5709 df-rn 5710 df-coss 38316 |
This theorem is referenced by: dmcoss2 38359 eldmcoss 38363 |
Copyright terms: Public domain | W3C validator |