Theorem relfld 5291

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 5283	. . . 4 |- ( Rel R -> R C_ ( dom R X. ran R ) )
2		uniss 3935	. . . 4 |- ( R C_ ( dom R X. ran R ) -> U. R C_ U. ( dom R X. ran R ) )
3		uniss 3935	. . . 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 4863	. . 3 |- U. U. ( dom R X. ran R ) C_ ( dom R u. ran R )
6	4, 5	sstrdi 3250	. 2 |- ( Rel R -> U. U. R C_ ( dom R u. ran R ) )
7		dmrnssfld 5020	. . 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 3255	1 |- ( Rel R -> U. U. R = ( dom R u. ran R ) )

Syntax hints:   -> wi 4   = wceq 1398   u. cun 3209   C_ wss 3211   U.cuni 3914   X. cxp 4747   dom cdm 4749   ran crn 4750   Rel wrel 4754

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 4228  ax-pow 4287  ax-pr 4322

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 2815  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-br 4110  df-opab 4172  df-xp 4755  df-rel 4756  df-cnv 4757  df-dm 4759  df-rn 4760

This theorem is referenced by:  relresfld  5292  relcoi1  5294  unidmrn  5295  relcnvfld  5296  unixpm  5298