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Theorem unidmrn 5074
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 4920 . . . 4 Rel 𝐴
2 relfld 5070 . . . 4 (Rel 𝐴 𝐴 = (dom 𝐴 ∪ ran 𝐴))
31, 2ax-mp 5 . . 3 𝐴 = (dom 𝐴 ∪ ran 𝐴)
43equncomi 3222 . 2 𝐴 = (ran 𝐴 ∪ dom 𝐴)
5 dfdm4 4734 . . 3 dom 𝐴 = ran 𝐴
6 df-rn 4553 . . 3 ran 𝐴 = dom 𝐴
75, 6uneq12i 3228 . 2 (dom 𝐴 ∪ ran 𝐴) = (ran 𝐴 ∪ dom 𝐴)
84, 7eqtr4i 2163 1 𝐴 = (dom 𝐴 ∪ ran 𝐴)
Colors of variables: wff set class
Syntax hints:   = wceq 1331  cun 3069   cuni 3739  ccnv 4541  dom cdm 4542  ran crn 4543  Rel wrel 4547
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4049  ax-pow 4101  ax-pr 4134
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3740  df-br 3933  df-opab 3993  df-xp 4548  df-rel 4549  df-cnv 4550  df-dm 4552  df-rn 4553
This theorem is referenced by:  relcnvfld  5075  dfdm2  5076
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