Theorem relfld 4419

Description: The double union of a relation is its field.

Assertion

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

Proof of Theorem relfld

Step	Hyp	Ref	Expression
1		relssdmrn 4416	. . . 4 |- (Rel R -> R C_ (dom R X. ran R))
2		uniss 3199	. . . 4 |- (R C_ (dom R X. ran R) -> U.R C_ U.(dom R X. ran R))
3		uniss 3199	. . . 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 24	. . 3 |- (Rel R -> U.U.R C_ U.U.(dom R X. ran R))
5		unixpss 4094	. . . 4 |- U.U.(dom R X. ran R) C_ (dom R u. ran R)
6	5	a1i 8	. . 3 |- (Rel R -> U.U.(dom R X. ran R) C_ (dom R u. ran R))
7	4, 6	sstrd 2627	. 2 |- (Rel R -> U.U.R C_ (dom R u. ran R))
8		dmrnssfld 4205	. . 3 |- (dom R u. ran R) C_ U.U.R
9	8	a1i 8	. 2 |- (Rel R -> (dom R u. ran R) C_ U.U.R)
10	7, 9	eqssd 2633	1 |- (Rel R -> U.U.R = (dom R u. ran R))

Syntax hints:   -> wi 3   = wceq 1298   u. cun 2591   C_ wss 2593  U.cuni 3177   X. cxp 3984  dom cdm 3986  ran crn 3987  Rel wrel 3991

This theorem is referenced by:  unidmrn 4420  unixp 4422  relcnvfld 14360  domrngref 14364  domfldref 14365  cnvref 14371  cnvref2 14372  cmprelid2 14447  resrelfld 14448  dispos 14632  fldleqt 15023  filnetlem1 15640

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-rex 2110  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-xp 4000  df-rel 4001  df-cnv 4002  df-dm 4004  df-rn 4005

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