Theorem relfld 3515

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 3513	. . . 4 |- (Rel R -> R (_ (dom R X. ran R))
2		uniss 2521	. . . 4 |- (R (_ (dom R X. ran R) -> U.R (_ U.(dom R X. ran R))
3		uniss 2521	. . . 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 3258	. . . 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 2074	. 2 |- (Rel R -> U.U.R (_ (dom R u. ran R))
8		dmrnssfld 3357	. . 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 2079	1 |- (Rel R -> U.U.R = (dom R u. ran R))

Syntax hints:   -> wi 3   = wceq 956   u. cun 2045   (_ wss 2047  U.cuni 2503   X. cxp 3168  dom cdm 3170  ran crn 3171  Rel wrel 3175

This theorem is referenced by:  unidmrn 3516  unixp 3517

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-pow 2742  ax-pr 2779

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-uni 2504  df-br 2620  df-opab 2667  df-xp 3184  df-rel 3185  df-cnv 3186  df-dm 3188  df-rn 3189

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