![]() |
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 6177 gives another proof). (Contributed by NM, 8-Apr-2007.) |
Ref | Expression |
---|---|
dmcnvcnv | ⊢ dom ◡◡𝐴 = dom 𝐴 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfdm4 5887 | . 2 ⊢ dom 𝐴 = ran ◡𝐴 | |
2 | df-rn 5680 | . 2 ⊢ ran ◡𝐴 = dom ◡◡𝐴 | |
3 | 1, 2 | eqtr2i 2760 | 1 ⊢ dom ◡◡𝐴 = dom 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1541 ◡ccnv 5668 dom cdm 5669 ran crn 5670 |
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 2702 ax-sep 5292 ax-nul 5299 ax-pr 5420 |
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 2709 df-cleq 2723 df-clel 2809 df-rab 3432 df-v 3475 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4523 df-sn 4623 df-pr 4625 df-op 4629 df-br 5142 df-opab 5204 df-cnv 5677 df-dm 5679 df-rn 5680 |
This theorem is referenced by: resdm2 6219 f1cnvcnv 6784 trrelsuperrel2dg 42193 |
Copyright terms: Public domain | W3C validator |