Theorem istsr2 17218

Description: The predicate is a toset. (Contributed by FL, 1-Nov-2009.) (Revised by Mario Carneiro, 22-Nov-2013.)

Hypothesis

Ref	Expression
istsr.1	|- X = dom R

Assertion

Ref	Expression
istsr2	|- ( R e. TosetRel <-> ( R e. PosetRel /\ A. x e. X A. y e. X ( x R y \/ y R x ) ) )

Distinct variable groups:   x, y, R   x, X, y

Proof of Theorem istsr2

Step	Hyp	Ref	Expression
1		istsr.1	. . 3 |- X = dom R
2	1	istsr 17217	. 2 |- ( R e. TosetRel <-> ( R e. PosetRel /\ ( X X. X ) C_ ( R u. `' R ) ) )
3		qfto 5517	. . 3 |- ( ( X X. X ) C_ ( R u. `' R ) <-> A. x e. X A. y e. X ( x R y \/ y R x ) )
4	3	anbi2i 730	. 2 |- ( ( R e. PosetRel /\ ( X X. X ) C_ ( R u. `' R ) ) <-> ( R e. PosetRel /\ A. x e. X A. y e. X ( x R y \/ y R x ) ) )
5	2, 4	bitri 264	1 |- ( R e. TosetRel <-> ( R e. PosetRel /\ A. x e. X A. y e. X ( x R y \/ y R x ) ) )

Syntax hints:   <-> wb 196   \/ wo 383   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   u. cun 3572   C_ wss 3574   class class class wbr 4653   X. cxp 5112   `'ccnv 5113   dom cdm 5114   PosetRelcps 17198   TosetRel ctsr 17199

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-br 4654  df-opab 4713  df-xp 5120  df-rel 5121  df-cnv 5122  df-dm 5124  df-tsr 17201

This theorem is referenced by:  tsrlin  17219  tsrss  17223