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Theorem unidmrn 4793
Description: The double union of the converse of a class is its field. (Contributed by NM, 4-Jun-2008.)
Assertion
Ref Expression
unidmrn  U. U. `'  dom  u.  ran

Proof of Theorem unidmrn
StepHypRef Expression
1 relcnv 4646 . . . 4  Rel  `'
2 relfld 4789 . . . 4  Rel  `'  U. U. `'  dom  `'  u.  ran  `'
31, 2ax-mp 7 . . 3  U. U. `'  dom  `'  u.  ran  `'
43equncomi 3083 . 2  U. U. `'  ran  `'  u.  dom  `'
5 dfdm4 4470 . . 3  dom  ran  `'
6 df-rn 4299 . . 3  ran  dom  `'
75, 6uneq12i 3089 . 2  dom  u.  ran  ran  `'  u.  dom  `'
84, 7eqtr4i 2060 1  U. U. `'  dom  u.  ran
Colors of variables: wff set class
Syntax hints:   wceq 1242    u. cun 2909   U.cuni 3571   `'ccnv 4287   dom cdm 4288   ran crn 4289   Rel wrel 4293
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-opab 3810  df-xp 4294  df-rel 4295  df-cnv 4296  df-dm 4298  df-rn 4299
This theorem is referenced by:  relcnvfld  4794  dfdm2  4795
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