Theorem relfld 5257

Description: The double union of a relation is its field. (Contributed by NM, 17-Sep-2006.)

Assertion

Ref	Expression
relfld	|- ( Rel R -> U. U. R = ( dom R u. ran R ) )

Proof of Theorem relfld

Step	Hyp	Ref	Expression
1		relssdmrn 5249	. . . 4 |- ( Rel R -> R C_ ( dom R X. ran R ) )
2		uniss 3909	. . . 4 |- ( R C_ ( dom R X. ran R ) -> U. R C_ U. ( dom R X. ran R ) )
3		uniss 3909	. . . 4 |- ( U. R C_ U. ( dom R X. ran R ) -> U. U. R C_ U. U. ( dom R X. ran R ) )
4	1, 2, 3	3syl 17	. . 3 |- ( Rel R -> U. U. R C_ U. U. ( dom R X. ran R ) )
5		unixpss 4832	. . 3 |- U. U. ( dom R X. ran R ) C_ ( dom R u. ran R )
6	4, 5	sstrdi 3236	. 2 |- ( Rel R -> U. U. R C_ ( dom R u. ran R ) )
7		dmrnssfld 4987	. . 3 |- ( dom R u. ran R ) C_ U. U. R
8	7	a1i 9	. 2 |- ( Rel R -> ( dom R u. ran R ) C_ U. U. R )
9	6, 8	eqssd 3241	1 |- ( Rel R -> U. U. R = ( dom R u. ran R ) )

Syntax hints:   -> wi 4   = wceq 1395   u. cun 3195   C_ wss 3197   U.cuni 3888   X. cxp 4717   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:  relresfld  5258  relcoi1  5260  unidmrn  5261  relcnvfld  5262  unixpm  5264