Theorem invsym2 14803

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 14801	. . . 4 |- ( ph -> ( Y N X ) C_ ( ( Y ( Hom ` C ) X ) X. ( X ( Hom ` C ) Y ) ) )
8		relxp 5045	. . . 4 |- Rel ( ( Y ( Hom ` C ) X ) X. ( X ( Hom ` C ) Y ) )
9		relss 5025	. . . 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 5304	. . 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 14802	. . . 4 |- ( ph -> ( f ( X N Y ) g <-> g ( Y N X ) f ) )
14		vex 3071	. . . . . 6 |- g e. _V
15		vex 3071	. . . . . 6 |- f e. _V
16	14, 15	brcnv 5120	. . . . 5 |- ( g `' ( X N Y ) f <-> f ( X N Y ) g )
17		df-br 4391	. . . . 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 4391	. . . 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 5037	. 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 1370   e. wcel 1758   C_ wss 3426   <.cop 3981   class class class wbr 4390   X. cxp 4936   `'ccnv 4937   Rel wrel 4943   ` cfv 5516  (class class class)co 6190   Basecbs 14276   Hom chom 14351   Catccat 14704  Invcinv 14786

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4501  ax-sep 4511  ax-nul 4519  ax-pow 4568  ax-pr 4629  ax-un 6472

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-reu 2802  df-rab 2804  df-v 3070  df-sbc 3285  df-csb 3387  df-dif 3429  df-un 3431  df-in 3433  df-ss 3440  df-nul 3736  df-if 3890  df-pw 3960  df-sn 3976  df-pr 3978  df-op 3982  df-uni 4190  df-iun 4271  df-br 4391  df-opab 4449  df-mpt 4450  df-id 4734  df-xp 4944  df-rel 4945  df-cnv 4946  df-co 4947  df-dm 4948  df-rn 4949  df-res 4950  df-ima 4951  df-iota 5479  df-fun 5518  df-fn 5519  df-f 5520  df-f1 5521  df-fo 5522  df-f1o 5523  df-fv 5524  df-ov 6193  df-oprab 6194  df-mpt2 6195  df-1st 6677  df-2nd 6678  df-sect 14788  df-inv 14789

This theorem is referenced by:  invf  14808  invf1o  14809  invinv  14810