Theorem domfldref 14365

Description: The domain of a reflexive relation is equal to its field .

Assertion

Ref	Expression
domfldref	|- ((Rel R /\ A.x e. U.U.RxRx) -> dom R = U.U.R)

Distinct variable group:   x,R

Proof of Theorem domfldref

Step	Hyp	Ref	Expression
1		domrngref 14364	. . . 4 |- ((Rel R /\ A.x e. U.U.RxRx) -> dom R = ran R)
2		eqtr 1904	. . . . . . 7 |- ((U.U.R = (dom R u. ran R) /\ (dom R u. ran R) = (dom R u. dom R)) -> U.U.R = (dom R u. dom R))
3		unidm 2743	. . . . . . 7 |- (dom R u. dom R) = dom R
4	2, 3	syl6req 1945	. . . . . 6 |- ((U.U.R = (dom R u. ran R) /\ (dom R u. ran R) = (dom R u. dom R)) -> dom R = U.U.R)
5	4	ex 402	. . . . 5 |- (U.U.R = (dom R u. ran R) -> ((dom R u. ran R) = (dom R u. dom R) -> dom R = U.U.R))
6		uneq2 2749	. . . . . 6 |- (dom R = ran R -> (dom R u. dom R) = (dom R u. ran R))
7	6	eqcomd 1889	. . . . 5 |- (dom R = ran R -> (dom R u. ran R) = (dom R u. dom R))
8	5, 7	syl5com 63	. . . 4 |- (dom R = ran R -> (U.U.R = (dom R u. ran R) -> dom R = U.U.R))
9	1, 8	syl 12	. . 3 |- ((Rel R /\ A.x e. U.U.RxRx) -> (U.U.R = (dom R u. ran R) -> dom R = U.U.R))
10		relfld 4419	. . 3 |- (Rel R -> U.U.R = (dom R u. ran R))
11	9, 10	syl5com 63	. 2 |- (Rel R -> ((Rel R /\ A.x e. U.U.RxRx) -> dom R = U.U.R))
12	11	anabsi5 553	1 |- ((Rel R /\ A.x e. U.U.RxRx) -> dom R = U.U.R)

Syntax hints:   -> wi 3   /\ wa 240   = wceq 1298  A.wral 2105   u. cun 2591  U.cuni 3177   class class class wbr 3338  dom cdm 3986  ran crn 3987  Rel wrel 3991

This theorem is referenced by:  preodom2 14567  dfps2 14634

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-3an 860  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-ral 2109  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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