Theorem relfld 5230

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 5222	. . . 4 |- ( Rel R -> R C_ ( dom R X. ran R ) )
2		uniss 3885	. . . 4 |- ( R C_ ( dom R X. ran R ) -> U. R C_ U. ( dom R X. ran R ) )
3		uniss 3885	. . . 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 4806	. . 3 |- U. U. ( dom R X. ran R ) C_ ( dom R u. ran R )
6	4, 5	sstrdi 3213	. 2 |- ( Rel R -> U. U. R C_ ( dom R u. ran R ) )
7		dmrnssfld 4960	. . 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 3218	1 |- ( Rel R -> U. U. R = ( dom R u. ran R ) )

Syntax hints:   -> wi 4   = wceq 1373   u. cun 3172   C_ wss 3174   U.cuni 3864   X. cxp 4691   dom cdm 4693   ran crn 4694   Rel wrel 4698

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-pow 4234  ax-pr 4269

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-rex 2492  df-v 2778  df-un 3178  df-in 3180  df-ss 3187  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-br 4060  df-opab 4122  df-xp 4699  df-rel 4700  df-cnv 4701  df-dm 4703  df-rn 4704

This theorem is referenced by:  relresfld  5231  relcoi1  5233  unidmrn  5234  relcnvfld  5235  unixpm  5237