Theorem relfld 5149

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 5141	. . . 4 |- ( Rel R -> R C_ ( dom R X. ran R ) )
2		uniss 3826	. . . 4 |- ( R C_ ( dom R X. ran R ) -> U. R C_ U. ( dom R X. ran R ) )
3		uniss 3826	. . . 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 4733	. . 3 |- U. U. ( dom R X. ran R ) C_ ( dom R u. ran R )
6	4, 5	sstrdi 3165	. 2 |- ( Rel R -> U. U. R C_ ( dom R u. ran R ) )
7		dmrnssfld 4883	. . 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 3170	1 |- ( Rel R -> U. U. R = ( dom R u. ran R ) )

Syntax hints:   -> wi 4   = wceq 1353   u. cun 3125   C_ wss 3127   U.cuni 3805   X. cxp 4618   dom cdm 4620   ran crn 4621   Rel wrel 4625

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 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-14 2149  ax-ext 2157  ax-sep 4116  ax-pow 4169  ax-pr 4203

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1459  df-sb 1761  df-eu 2027  df-mo 2028  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ral 2458  df-rex 2459  df-v 2737  df-un 3131  df-in 3133  df-ss 3140  df-pw 3574  df-sn 3595  df-pr 3596  df-op 3598  df-uni 3806  df-br 3999  df-opab 4060  df-xp 4626  df-rel 4627  df-cnv 4628  df-dm 4630  df-rn 4631

This theorem is referenced by:  relresfld  5150  relcoi1  5152  unidmrn  5153  relcnvfld  5154  unixpm  5156