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Theorem unidmrn 6103
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 5940 . . . 4 Rel 𝐴
2 relfld 6099 . . . 4 (Rel 𝐴 𝐴 = (dom 𝐴 ∪ ran 𝐴))
31, 2ax-mp 5 . . 3 𝐴 = (dom 𝐴 ∪ ran 𝐴)
43equncomi 4107 . 2 𝐴 = (ran 𝐴 ∪ dom 𝐴)
5 dfdm4 5737 . . 3 dom 𝐴 = ran 𝐴
6 df-rn 5539 . . 3 ran 𝐴 = dom 𝐴
75, 6uneq12i 4113 . 2 (dom 𝐴 ∪ ran 𝐴) = (ran 𝐴 ∪ dom 𝐴)
84, 7eqtr4i 2847 1 𝐴 = (dom 𝐴 ∪ ran 𝐴)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1538  cun 3908   cuni 4811  ccnv 5527  dom cdm 5528  ran crn 5529  Rel wrel 5533
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793  ax-sep 5176  ax-nul 5183  ax-pr 5303
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2623  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ral 3131  df-rex 3132  df-rab 3135  df-v 3473  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4267  df-if 4441  df-pw 4514  df-sn 4541  df-pr 4543  df-op 4547  df-uni 4812  df-br 5040  df-opab 5102  df-xp 5534  df-rel 5535  df-cnv 5536  df-dm 5538  df-rn 5539
This theorem is referenced by:  relcnvfld  6104  dfdm2  6105
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