Theorem psrn 9993

Description: The range of a poset equals it domain.

Hypothesis

Ref	Expression
psref.1	|- X = dom R

Assertion

Ref	Expression
psrn	|- (R e. Poset -> X = ran R)

Proof of Theorem psrn

Step	Hyp	Ref	Expression
1		psdmrn 9991	. . 3 |- (R e. Poset -> (dom R = U.U.R /\ ran R = U.U.R))
2		eqtr3 1907	. . 3 |- ((dom R = U.U.R /\ ran R = U.U.R) -> dom R = ran R)
3	1, 2	syl 12	. 2 |- (R e. Poset -> dom R = ran R)
4		psref.1	. 2 |- X = dom R
5	3, 4	syl5eq 1940	1 |- (R e. Poset -> X = ran R)

Syntax hints:   -> wi 3   /\ wa 240   = wceq 1298   e. wcel 1300  U.cuni 3177  dom cdm 3986  ran crn 3987  Posetcps 9980

This theorem is referenced by:  spwpr4 10006  spwpr4c 10009

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-id 3586  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ps 9984

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