|   | Metamath Proof Explorer | < Previous  
      Next > Nearby theorems | |
| Mirrors > Home > MPE Home > Th. List > dmcnvcnv | Structured version Visualization version GIF version | ||
| Description: The domain of the double converse of a class is equal to its domain (even when that class in not a relation, in which case dfrel2 6208 gives another proof). (Contributed by NM, 8-Apr-2007.) | 
| Ref | Expression | 
|---|---|
| dmcnvcnv | ⊢ dom ◡◡𝐴 = dom 𝐴 | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | dfdm4 5905 | . 2 ⊢ dom 𝐴 = ran ◡𝐴 | |
| 2 | df-rn 5695 | . 2 ⊢ ran ◡𝐴 = dom ◡◡𝐴 | |
| 3 | 1, 2 | eqtr2i 2765 | 1 ⊢ dom ◡◡𝐴 = dom 𝐴 | 
| Colors of variables: wff setvar class | 
| Syntax hints: = wceq 1539 ◡ccnv 5683 dom cdm 5684 ran crn 5685 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pr 5431 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-br 5143 df-opab 5205 df-cnv 5692 df-dm 5694 df-rn 5695 | 
| This theorem is referenced by: resdm2 6250 f1cnvcnv 6812 trrelsuperrel2dg 43689 | 
| Copyright terms: Public domain | W3C validator |