![]() |
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 6188 gives another proof). (Contributed by NM, 8-Apr-2007.) |
Ref | Expression |
---|---|
dmcnvcnv | ⊢ dom ◡◡𝐴 = dom 𝐴 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfdm4 5895 | . 2 ⊢ dom 𝐴 = ran ◡𝐴 | |
2 | df-rn 5687 | . 2 ⊢ ran ◡𝐴 = dom ◡◡𝐴 | |
3 | 1, 2 | eqtr2i 2761 | 1 ⊢ dom ◡◡𝐴 = dom 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1541 ◡ccnv 5675 dom cdm 5676 ran crn 5677 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2703 ax-sep 5299 ax-nul 5306 ax-pr 5427 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-sb 2068 df-clab 2710 df-cleq 2724 df-clel 2810 df-rab 3433 df-v 3476 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-br 5149 df-opab 5211 df-cnv 5684 df-dm 5686 df-rn 5687 |
This theorem is referenced by: resdm2 6230 f1cnvcnv 6797 trrelsuperrel2dg 42412 |
Copyright terms: Public domain | W3C validator |