Theorem relfld 4846

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 4841	. . . 4 |- ( Rel R -> R C_ ( dom R X. ran R ) )
2		uniss 3601	. . . 4 |- ( R C_ ( dom R X. ran R ) -> U. R C_ U. ( dom R X. ran R ) )
3		uniss 3601	. . . 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 4451	. . 3 |- U. U. ( dom R X. ran R ) C_ ( dom R u. ran R )
6	4, 5	syl6ss 2957	. 2 |- ( Rel R -> U. U. R C_ ( dom R u. ran R ) )
7		dmrnssfld 4595	. . 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 2962	1 |- ( Rel R -> U. U. R = ( dom R u. ran R ) )

Syntax hints:   -> wi 4   = wceq 1243   u. cun 2915   C_ wss 2917   U.cuni 3580   X. cxp 4343   dom cdm 4345   ran crn 4346   Rel wrel 4350

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-opab 3819  df-xp 4351  df-rel 4352  df-cnv 4353  df-dm 4355  df-rn 4356

This theorem is referenced by:  relresfld  4847  relcoi1  4849  unidmrn  4850  relcnvfld  4851  unixpm  4853