Theorem invsym2 15668

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
invsym2	|- ( ph -> `' ( X N Y ) = ( Y N X ) )

Proof of Theorem invsym2

Dummy variables f g are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		invfval.b	. . . . 5 |- B = ( Base ` C )
2		invfval.n	. . . . 5 |- N = (Inv ` C )
3		invfval.c	. . . . 5 |- ( ph -> C e. Cat )
4		invfval.y	. . . . 5 |- ( ph -> Y e. B )
5		invfval.x	. . . . 5 |- ( ph -> X e. B )
6		eqid 2451	. . . . 5 |- ( Hom ` C ) = ( Hom ` C )
7	1, 2, 3, 4, 5, 6	invss 15666	. . . 4 |- ( ph -> ( Y N X ) C_ ( ( Y ( Hom ` C ) X ) X. ( X ( Hom ` C ) Y ) ) )
8		relxp 4942	. . . 4 |- Rel ( ( Y ( Hom ` C ) X ) X. ( X ( Hom ` C ) Y ) )
9		relss 4922	. . . 4 |- ( ( Y N X ) C_ ( ( Y ( Hom ` C ) X ) X. ( X ( Hom ` C ) Y ) ) -> ( Rel ( ( Y ( Hom ` C ) X ) X. ( X ( Hom ` C ) Y ) ) -> Rel ( Y N X ) ) )
10	7, 8, 9	mpisyl 21	. . 3 |- ( ph -> Rel ( Y N X ) )
11		relcnv 5207	. . 3 |- Rel `' ( X N Y )
12	10, 11	jctil 540	. 2 |- ( ph -> ( Rel `' ( X N Y ) /\ Rel ( Y N X ) ) )
13	1, 2, 3, 5, 4	invsym 15667	. . . 4 |- ( ph -> ( f ( X N Y ) g <-> g ( Y N X ) f ) )
14		vex 3048	. . . . . 6 |- g e. _V
15		vex 3048	. . . . . 6 |- f e. _V
16	14, 15	brcnv 5017	. . . . 5 |- ( g `' ( X N Y ) f <-> f ( X N Y ) g )
17		df-br 4403	. . . . 5 |- ( g `' ( X N Y ) f <-> <. g , f >. e. `' ( X N Y ) )
18	16, 17	bitr3i 255	. . . 4 |- ( f ( X N Y ) g <-> <. g , f >. e. `' ( X N Y ) )
19		df-br 4403	. . . 4 |- ( g ( Y N X ) f <-> <. g , f >. e. ( Y N X ) )
20	13, 18, 19	3bitr3g 291	. . 3 |- ( ph -> ( <. g , f >. e. `' ( X N Y ) <-> <. g , f >. e. ( Y N X ) ) )
21	20	eqrelrdv2 4934	. 2 |- ( ( ( Rel `' ( X N Y ) /\ Rel ( Y N X ) ) /\ ph ) -> `' ( X N Y ) = ( Y N X ) )
22	12, 21	mpancom 675	1 |- ( ph -> `' ( X N Y ) = ( Y N X ) )

Syntax hints:   -> wi 4   /\ wa 371   = wceq 1444   e. wcel 1887   C_ wss 3404   <.cop 3974   class class class wbr 4402   X. cxp 4832   `'ccnv 4833   Rel wrel 4839   ` cfv 5582  (class class class)co 6290   Basecbs 15121   Hom chom 15201   Catccat 15570  Invcinv 15650

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-rex 2743  df-reu 2744  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-op 3975  df-uni 4199  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-id 4749  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-1st 6793  df-2nd 6794  df-sect 15652  df-inv 15653

This theorem is referenced by:  invf  15673  invf1o  15674  invinv  15675  cicsym  15709