Theorem relcnvfld 5262

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 5257	. 2 |- ( Rel R -> U. U. R = ( dom R u. ran R ) )
2		unidmrn 5261	. 2 |- U. U. `' R = ( dom R u. ran R )
3	1, 2	eqtr4di 2280	1 |- ( Rel R -> U. U. R = U. U. `' R )

Syntax hints:   -> wi 4   = wceq 1395   u. cun 3195   U.cuni 3888   `'ccnv 4718   dom cdm 4719   ran crn 4720   Rel wrel 4724

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-br 4084  df-opab 4146  df-xp 4725  df-rel 4726  df-cnv 4727  df-dm 4729  df-rn 4730

This theorem is referenced by: (None)