Theorem tpostpos2 7373

Description: Value of the double transposition for a relation on triples. (Contributed by Mario Carneiro, 16-Sep-2015.)

Assertion

Ref	Expression
tpostpos2	|- ( ( Rel F /\ Rel dom F ) -> tpos tpos F = F )

Proof of Theorem tpostpos2

Step	Hyp	Ref	Expression
1		tpostpos 7372	. 2 |- tpos tpos F = ( F i^i ( ( ( _V X. _V ) u. { (/) } ) X. _V ) )
2		relrelss 5659	. . . 4 |- ( ( Rel F /\ Rel dom F ) <-> F C_ ( ( _V X. _V ) X. _V ) )
3		ssun1 3776	. . . . . 6 |- ( _V X. _V ) C_ ( ( _V X. _V ) u. { (/) } )
4		xpss1 5228	. . . . . 6 |- ( ( _V X. _V ) C_ ( ( _V X. _V ) u. { (/) } ) -> ( ( _V X. _V ) X. _V ) C_ ( ( ( _V X. _V ) u. { (/) } ) X. _V ) )
5	3, 4	ax-mp 5	. . . . 5 |- ( ( _V X. _V ) X. _V ) C_ ( ( ( _V X. _V ) u. { (/) } ) X. _V )
6		sstr 3611	. . . . 5 |- ( ( F C_ ( ( _V X. _V ) X. _V ) /\ ( ( _V X. _V ) X. _V ) C_ ( ( ( _V X. _V ) u. { (/) } ) X. _V ) ) -> F C_ ( ( ( _V X. _V ) u. { (/) } ) X. _V ) )
7	5, 6	mpan2 707	. . . 4 |- ( F C_ ( ( _V X. _V ) X. _V ) -> F C_ ( ( ( _V X. _V ) u. { (/) } ) X. _V ) )
8	2, 7	sylbi 207	. . 3 |- ( ( Rel F /\ Rel dom F ) -> F C_ ( ( ( _V X. _V ) u. { (/) } ) X. _V ) )
9		df-ss 3588	. . 3 |- ( F C_ ( ( ( _V X. _V ) u. { (/) } ) X. _V ) <-> ( F i^i ( ( ( _V X. _V ) u. { (/) } ) X. _V ) ) = F )
10	8, 9	sylib 208	. 2 |- ( ( Rel F /\ Rel dom F ) -> ( F i^i ( ( ( _V X. _V ) u. { (/) } ) X. _V ) ) = F )
11	1, 10	syl5eq 2668	1 |- ( ( Rel F /\ Rel dom F ) -> tpos tpos F = F )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   _Vcvv 3200   u. cun 3572   i^i cin 3573   C_ wss 3574   (/)c0 3915   {csn 4177   X. cxp 5112   dom cdm 5114   Rel wrel 5119  tpos ctpos 7351

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-8 1992  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-pow 4843  ax-pr 4906  ax-un 6949

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-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  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-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-fv 5896  df-tpos 7352

This theorem is referenced by:  2oppchomf  16384  mattpostpos  20260