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Theorem unidmrn 6232
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 6057 . . . 4 Rel 𝐴
2 relfld 6228 . . . 4 (Rel 𝐴 𝐴 = (dom 𝐴 ∪ ran 𝐴))
31, 2ax-mp 5 . . 3 𝐴 = (dom 𝐴 ∪ ran 𝐴)
43equncomi 4116 . 2 𝐴 = (ran 𝐴 ∪ dom 𝐴)
5 dfdm4 5852 . . 3 dom 𝐴 = ran 𝐴
6 df-rn 5645 . . 3 ran 𝐴 = dom 𝐴
75, 6uneq12i 4122 . 2 (dom 𝐴 ∪ ran 𝐴) = (ran 𝐴 ∪ dom 𝐴)
84, 7eqtr4i 2764 1 𝐴 = (dom 𝐴 ∪ ran 𝐴)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1542  cun 3909   cuni 4866  ccnv 5633  dom cdm 5634  ran crn 5635  Rel wrel 5639
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-ral 3062  df-rex 3071  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-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-xp 5640  df-rel 5641  df-cnv 5642  df-dm 5644  df-rn 5645
This theorem is referenced by:  relcnvfld  6233  dfdm2  6234
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