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Theorem unidmrn 4950
Description: The double union of the converse of a class is its field. (Contributed by NM, 4-Jun-2008.)
Assertion
Ref Expression
unidmrn 𝐴 = (dom 𝐴 ∪ ran 𝐴)

Proof of Theorem unidmrn
StepHypRef Expression
1 relcnv 4797 . . . 4 Rel 𝐴
2 relfld 4946 . . . 4 (Rel 𝐴 𝐴 = (dom 𝐴 ∪ ran 𝐴))
31, 2ax-mp 7 . . 3 𝐴 = (dom 𝐴 ∪ ran 𝐴)
43equncomi 3144 . 2 𝐴 = (ran 𝐴 ∪ dom 𝐴)
5 dfdm4 4616 . . 3 dom 𝐴 = ran 𝐴
6 df-rn 4439 . . 3 ran 𝐴 = dom 𝐴
75, 6uneq12i 3150 . 2 (dom 𝐴 ∪ ran 𝐴) = (ran 𝐴 ∪ dom 𝐴)
84, 7eqtr4i 2111 1 𝐴 = (dom 𝐴 ∪ ran 𝐴)
Colors of variables: wff set class
Syntax hints:   = wceq 1289  cun 2995   cuni 3648  ccnv 4427  dom cdm 4428  ran crn 4429  Rel wrel 4433
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-pr 4027
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-br 3838  df-opab 3892  df-xp 4434  df-rel 4435  df-cnv 4436  df-dm 4438  df-rn 4439
This theorem is referenced by:  relcnvfld  4951  dfdm2  4952
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