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

Theorem unidmrn 5399
Description: The double union of the converse of a class is its field. (Contributed by NM, 4-Jun-2008.)
Assertion
Ref Expression
unidmrn  |-  U. U. `' A  =  ( dom  A  u.  ran  A
)

Proof of Theorem unidmrn
StepHypRef Expression
1 relcnv 5242 . . . 4  |-  Rel  `' A
2 relfld 5395 . . . 4  |-  ( Rel  `' A  ->  U. U. `' A  =  ( dom  `' A  u.  ran  `' A ) )
31, 2ax-mp 8 . . 3  |-  U. U. `' A  =  ( dom  `' A  u.  ran  `' A )
43equncomi 3493 . 2  |-  U. U. `' A  =  ( ran  `' A  u.  dom  `' A )
5 dfdm4 5063 . . 3  |-  dom  A  =  ran  `' A
6 df-rn 4889 . . 3  |-  ran  A  =  dom  `' A
75, 6uneq12i 3499 . 2  |-  ( dom 
A  u.  ran  A
)  =  ( ran  `' A  u.  dom  `' A )
84, 7eqtr4i 2459 1  |-  U. U. `' A  =  ( dom  A  u.  ran  A
)
Colors of variables: wff set class
Syntax hints:    = wceq 1652    u. cun 3318   U.cuni 4015   `'ccnv 4877   dom cdm 4878   ran crn 4879   Rel wrel 4883
This theorem is referenced by:  relcnvfld  5400  dfdm2  5401
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pr 4403
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-xp 4884  df-rel 4885  df-cnv 4886  df-dm 4888  df-rn 4889
  Copyright terms: Public domain W3C validator