Theorem invsym2 15014

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 2467	. . . . 5 |- ( Hom ` C ) = ( Hom ` C )
7	1, 2, 3, 4, 5, 6	invss 15012	. . . 4 |- ( ph -> ( Y N X ) C_ ( ( Y ( Hom ` C ) X ) X. ( X ( Hom ` C ) Y ) ) )
8		relxp 5108	. . . 4 |- Rel ( ( Y ( Hom ` C ) X ) X. ( X ( Hom ` C ) Y ) )
9		relss 5088	. . . 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 18	. . 3 |- ( ph -> Rel ( Y N X ) )
11		relcnv 5372	. . 3 |- Rel `' ( X N Y )
12	10, 11	jctil 537	. 2 |- ( ph -> ( Rel `' ( X N Y ) /\ Rel ( Y N X ) ) )
13	1, 2, 3, 5, 4	invsym 15013	. . . 4 |- ( ph -> ( f ( X N Y ) g <-> g ( Y N X ) f ) )
14		vex 3116	. . . . . 6 |- g e. _V
15		vex 3116	. . . . . 6 |- f e. _V
16	14, 15	brcnv 5183	. . . . 5 |- ( g `' ( X N Y ) f <-> f ( X N Y ) g )
17		df-br 4448	. . . . 5 |- ( g `' ( X N Y ) f <-> <. g , f >. e. `' ( X N Y ) )
18	16, 17	bitr3i 251	. . . 4 |- ( f ( X N Y ) g <-> <. g , f >. e. `' ( X N Y ) )
19		df-br 4448	. . . 4 |- ( g ( Y N X ) f <-> <. g , f >. e. ( Y N X ) )
20	13, 18, 19	3bitr3g 287	. . 3 |- ( ph -> ( <. g , f >. e. `' ( X N Y ) <-> <. g , f >. e. ( Y N X ) ) )
21	20	eqrelrdv2 5100	. 2 |- ( ( ( Rel `' ( X N Y ) /\ Rel ( Y N X ) ) /\ ph ) -> `' ( X N Y ) = ( Y N X ) )
22	12, 21	mpancom 669	1 |- ( ph -> `' ( X N Y ) = ( Y N X ) )

Syntax hints:   -> wi 4   /\ wa 369   = wceq 1379   e. wcel 1767   C_ wss 3476   <.cop 4033   class class class wbr 4447   X. cxp 4997   `'ccnv 4998   Rel wrel 5004   ` cfv 5586  (class class class)co 6282   Basecbs 14486   Hom chom 14562   Catccat 14915  Invcinv 14997

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-1st 6781  df-2nd 6782  df-sect 14999  df-inv 15000

This theorem is referenced by:  invf  15019  invf1o  15020  invinv  15021