Theorem cnvtrrel 37962

Description: The converse of a transitive relation is a transitive relation. (Contributed by Richard Penner, 25-Dec-2019.)

Assertion

Ref	Expression
cnvtrrel	|- ( ( S o. S ) C_ S <-> ( `' S o. `' S ) C_ `' S )

Proof of Theorem cnvtrrel

Step	Hyp	Ref	Expression
1		cnvss 5294	. . 3 |- ( ( S o. S ) C_ S -> `' ( S o. S ) C_ `' S )
2		cnvss 5294	. . . 4 |- ( `' ( S o. S ) C_ `' S -> `' `' ( S o. S ) C_ `' `' S )
3		cnvco 5308	. . . . . . . . 9 |- `' ( S o. S ) = ( `' S o. `' S )
4	3	cnveqi 5297	. . . . . . . 8 |- `' `' ( S o. S ) = `' ( `' S o. `' S )
5		cnvco 5308	. . . . . . . 8 |- `' ( `' S o. `' S ) = ( `' `' S o. `' `' S )
6		cocnvcnv1 5646	. . . . . . . . 9 |- ( `' `' S o. `' `' S ) = ( S o. `' `' S )
7		cocnvcnv2 5647	. . . . . . . . 9 |- ( S o. `' `' S ) = ( S o. S )
8	6, 7	eqtri 2644	. . . . . . . 8 |- ( `' `' S o. `' `' S ) = ( S o. S )
9	4, 5, 8	3eqtri 2648	. . . . . . 7 |- `' `' ( S o. S ) = ( S o. S )
10	9	sseq1i 3629	. . . . . 6 |- ( `' `' ( S o. S ) C_ `' `' S <-> ( S o. S ) C_ `' `' S )
11	10	biimpi 206	. . . . 5 |- ( `' `' ( S o. S ) C_ `' `' S -> ( S o. S ) C_ `' `' S )
12		cnvcnvss 5589	. . . . 5 |- `' `' S C_ S
13	11, 12	syl6ss 3615	. . . 4 |- ( `' `' ( S o. S ) C_ `' `' S -> ( S o. S ) C_ S )
14	2, 13	syl 17	. . 3 |- ( `' ( S o. S ) C_ `' S -> ( S o. S ) C_ S )
15	1, 14	impbii 199	. 2 |- ( ( S o. S ) C_ S <-> `' ( S o. S ) C_ `' S )
16	3	sseq1i 3629	. 2 |- ( `' ( S o. S ) C_ `' S <-> ( `' S o. `' S ) C_ `' S )
17	15, 16	bitri 264	1 |- ( ( S o. S ) C_ S <-> ( `' S o. `' S ) C_ `' S )

Syntax hints:   <-> wb 196   C_ wss 3574   `'ccnv 5113   o. ccom 5118

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-co 5123  df-dm 5124  df-rn 5125  df-res 5126

This theorem is referenced by: (None)