MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  unidmrn Structured version   Visualization version   GIF version

Theorem unidmrn 6286
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 6111 . . . 4 Rel 𝐴
2 relfld 6282 . . . 4 (Rel 𝐴 𝐴 = (dom 𝐴 ∪ ran 𝐴))
31, 2ax-mp 5 . . 3 𝐴 = (dom 𝐴 ∪ ran 𝐴)
43equncomi 4154 . 2 𝐴 = (ran 𝐴 ∪ dom 𝐴)
5 dfdm4 5900 . . 3 dom 𝐴 = ran 𝐴
6 df-rn 5691 . . 3 ran 𝐴 = dom 𝐴
75, 6uneq12i 4160 . 2 (dom 𝐴 ∪ ran 𝐴) = (ran 𝐴 ∪ dom 𝐴)
84, 7eqtr4i 2758 1 𝐴 = (dom 𝐴 ∪ ran 𝐴)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1533  cun 3945   cuni 4910  ccnv 5679  dom cdm 5680  ran crn 5681  Rel wrel 5685
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2698  ax-sep 5301  ax-nul 5308  ax-pr 5431
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2705  df-cleq 2719  df-clel 2805  df-ral 3058  df-rex 3067  df-rab 3429  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4325  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4911  df-br 5151  df-opab 5213  df-xp 5686  df-rel 5687  df-cnv 5688  df-dm 5690  df-rn 5691
This theorem is referenced by:  relcnvfld  6287  dfdm2  6288
  Copyright terms: Public domain W3C validator