Theorem invsym 16422

Description: The inverse relation is symmetric. (Contributed by Mario Carneiro, 2-Jan-2017.)

Hypotheses

Ref	Expression
invfval.b	|- B = ( Base ` C )
invfval.n	|- N = (Inv ` C )
invfval.c	|- ( ph -> C e. Cat )
invfval.x	|- ( ph -> X e. B )
invfval.y	|- ( ph -> Y e. B )

Assertion

Ref	Expression
invsym	|- ( ph -> ( F ( X N Y ) G <-> G ( Y N X ) F ) )

Proof of Theorem invsym

Step	Hyp	Ref	Expression
1		invfval.b	. . . 4 |- B = ( Base ` C )
2		invfval.n	. . . 4 |- N = (Inv ` C )
3		invfval.c	. . . 4 |- ( ph -> C e. Cat )
4		invfval.x	. . . 4 |- ( ph -> X e. B )
5		invfval.y	. . . 4 |- ( ph -> Y e. B )
6		eqid 2622	. . . 4 |- (Sect ` C ) = (Sect ` C )
7	1, 2, 3, 4, 5, 6	isinv 16420	. . 3 |- ( ph -> ( F ( X N Y ) G <-> ( F ( X (Sect ` C ) Y ) G /\ G ( Y (Sect ` C ) X ) F ) ) )
8		ancom 466	. . 3 |- ( ( F ( X (Sect ` C ) Y ) G /\ G ( Y (Sect ` C ) X ) F ) <-> ( G ( Y (Sect ` C ) X ) F /\ F ( X (Sect ` C ) Y ) G ) )
9	7, 8	syl6bb 276	. 2 |- ( ph -> ( F ( X N Y ) G <-> ( G ( Y (Sect ` C ) X ) F /\ F ( X (Sect ` C ) Y ) G ) ) )
10	1, 2, 3, 5, 4, 6	isinv 16420	. 2 |- ( ph -> ( G ( Y N X ) F <-> ( G ( Y (Sect ` C ) X ) F /\ F ( X (Sect ` C ) Y ) G ) ) )
11	9, 10	bitr4d 271	1 |- ( ph -> ( F ( X N Y ) G <-> G ( Y N X ) F ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   Basecbs 15857   Catccat 16325  Sectcsect 16404  Invcinv 16405

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-rep 4771  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-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  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-iun 4522  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-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-sect 16407  df-inv 16408

This theorem is referenced by:  invsym2  16423  inviso2  16427  invisoinvl  16450  invisoinvr  16451