Theorem psrn 17209

Description: The range of a poset equals it domain. (Contributed by NM, 7-Jul-2008.)

Hypothesis

Ref	Expression
psref.1	|- X = dom R

Assertion

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

Proof of Theorem psrn

Step	Hyp	Ref	Expression
1		psref.1	. 2 |- X = dom R
2		psdmrn 17207	. . 3 |- ( R e. PosetRel -> ( dom R = U. U. R /\ ran R = U. U. R ) )
3		eqtr3 2643	. . 3 |- ( ( dom R = U. U. R /\ ran R = U. U. R ) -> dom R = ran R )
4	2, 3	syl 17	. 2 |- ( R e. PosetRel -> dom R = ran R )
5	1, 4	syl5eq 2668	1 |- ( R e. PosetRel -> X = ran R )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   U.cuni 4436   dom cdm 5114   ran crn 5115   PosetRelcps 17198

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ps 17200

This theorem is referenced by:  cnvtsr  17222  ordtbas2  20995  ordtcnv  21005  ordtrest2  21008