Theorem relcnvfld 5272

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 5267	. 2 |- ( Rel R -> U. U. R = ( dom R u. ran R ) )
2		unidmrn 5271	. 2 |- U. U. `' R = ( dom R u. ran R )
3	1, 2	eqtr4di 2281	1 |- ( Rel R -> U. U. R = U. U. `' R )

Syntax hints:   -> wi 4   = wceq 1397   u. cun 3197   U.cuni 3894   `'ccnv 4726   dom cdm 4727   ran crn 4728   Rel wrel 4732

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2204  ax-ext 2212  ax-sep 4208  ax-pow 4266  ax-pr 4301

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1810  df-eu 2081  df-mo 2082  df-clab 2217  df-cleq 2223  df-clel 2226  df-nfc 2362  df-ral 2514  df-rex 2515  df-v 2803  df-un 3203  df-in 3205  df-ss 3212  df-pw 3655  df-sn 3676  df-pr 3677  df-op 3679  df-uni 3895  df-br 4090  df-opab 4152  df-xp 4733  df-rel 4734  df-cnv 4735  df-dm 4737  df-rn 4738

This theorem is referenced by: (None)