Theorem istsr 17217

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
istsr	|- ( R e. TosetRel <-> ( R e. PosetRel /\ ( X X. X ) C_ ( R u. `' R ) ) )

Proof of Theorem istsr

Dummy variable r is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dmeq 5324	. . . . 5 |- ( r = R -> dom r = dom R )
2		istsr.1	. . . . 5 |- X = dom R
3	1, 2	syl6eqr 2674	. . . 4 |- ( r = R -> dom r = X )
4	3	sqxpeqd 5141	. . 3 |- ( r = R -> ( dom r X. dom r ) = ( X X. X ) )
5		id 22	. . . 4 |- ( r = R -> r = R )
6		cnveq 5296	. . . 4 |- ( r = R -> `' r = `' R )
7	5, 6	uneq12d 3768	. . 3 |- ( r = R -> ( r u. `' r ) = ( R u. `' R ) )
8	4, 7	sseq12d 3634	. 2 |- ( r = R -> ( ( dom r X. dom r ) C_ ( r u. `' r ) <-> ( X X. X ) C_ ( R u. `' R ) ) )
9		df-tsr 17201	. 2 |- TosetRel = { r e. PosetRel | ( dom r X. dom r ) C_ ( r u. `' r ) }
10	8, 9	elrab2 3366	1 |- ( R e. TosetRel <-> ( R e. PosetRel /\ ( X X. X ) C_ ( R u. `' R ) ) )

Syntax hints:   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   u. cun 3572   C_ wss 3574   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

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-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  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-cnv 5122  df-dm 5124  df-tsr 17201

This theorem is referenced by:  istsr2  17218  tsrlemax  17220  tsrps  17221  cnvtsr  17222  letsr  17227  tsrdir  17238