Theorem relcnvfld 5295

Description: if R is a relation, its double union equals the double union of its converse. (Contributed by FL, 5-Jan-2009.)

Assertion

Ref	Expression
relcnvfld	|- ( Rel R -> U. U. R = U. U. `' R )

Proof of Theorem relcnvfld

Step	Hyp	Ref	Expression
1		relfld 5290	. 2 |- ( Rel R -> U. U. R = ( dom R u. ran R ) )
2		unidmrn 5294	. 2 |- U. U. `' R = ( dom R u. ran R )
3	1, 2	eqtr4di 2283	1 |- ( Rel R -> U. U. R = U. U. `' R )

Syntax hints:   -> wi 4   = wceq 1398   u. cun 3208   U.cuni 3913   `'ccnv 4747   dom cdm 4748   ran crn 4749   Rel wrel 4753

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2206  ax-ext 2214  ax-sep 4227  ax-pow 4286  ax-pr 4321

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-v 2814  df-un 3214  df-in 3216  df-ss 3223  df-pw 3670  df-sn 3694  df-pr 3695  df-op 3697  df-uni 3914  df-br 4109  df-opab 4171  df-xp 4754  df-rel 4755  df-cnv 4756  df-dm 4758  df-rn 4759

This theorem is referenced by: (None)