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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 6142 gives another proof). (Contributed by NM, 8-Apr-2007.) |
Ref | Expression |
---|---|
dmcnvcnv | ⊢ dom ◡◡𝐴 = dom 𝐴 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfdm4 5852 | . 2 ⊢ dom 𝐴 = ran ◡𝐴 | |
2 | df-rn 5645 | . 2 ⊢ ran ◡𝐴 = dom ◡◡𝐴 | |
3 | 1, 2 | eqtr2i 2762 | 1 ⊢ dom ◡◡𝐴 = dom 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1542 ◡ccnv 5633 dom cdm 5634 ran crn 5635 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 ax-sep 5257 ax-nul 5264 ax-pr 5385 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-rab 3407 df-v 3446 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-sn 4588 df-pr 4590 df-op 4594 df-br 5107 df-opab 5169 df-cnv 5642 df-dm 5644 df-rn 5645 |
This theorem is referenced by: resdm2 6184 f1cnvcnv 6749 trrelsuperrel2dg 42031 |
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