Theorem relfld 3507

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		relssdr 3505	. . . 4 |- (Rel R -> R (_ (dom R X. ran R))
2		uniss 2516	. . . 4 |- (R (_ (dom R X. ran R) -> U.R (_ U.(dom R X. ran R))
3		uniss 2516	. . . 4 |- (U.R (_ U.(dom R X. ran R) -> U.U.R (_ U.U.(dom R X. ran R))
4	1, 2, 3	3syl 20	. . 3 |- (Rel R -> U.U.R (_ U.U.(dom R X. ran R))
5		unixpss 3253	. . . 4 |- U.U.(dom R X. ran R) (_ (dom R u. ran R)
6	5	a1i 8	. . 3 |- (Rel R -> U.U.(dom R X. ran R) (_ (dom R u. ran R))
7	4, 6	sstrd 2070	. 2 |- (Rel R -> U.U.R (_ (dom R u. ran R))
8		dmrnssfld 3351	. . 3 |- (dom R u. ran R) (_ U.U.R
9	8	a1i 8	. 2 |- (Rel R -> (dom R u. ran R) (_ U.U.R)
10	7, 9	eqssd 2075	1 |- (Rel R -> U.U.R = (dom R u. ran R))

Syntax hints:   -> wi 3   = wceq 954   u. cun 2041   (_ wss 2043  U.cuni 2498   X. cxp 3163  dom cdm 3165  ran crn 3166  Rel wrel 3170

This theorem is referenced by:  unidmrn 3508  unixp 3509

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-10 964  ax-11 965  ax-12 966  ax-13 967  ax-14 968  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457  ax-sep 2698  ax-pow 2737  ax-pr 2774

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-v 1808  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-nul 2277  df-pw 2398  df-sn 2408  df-pr 2409  df-op 2412  df-uni 2499  df-br 2615  df-opab 2662  df-xp 3179  df-rel 3180  df-cnv 3181  df-dm 3183  df-rn 3184

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