Theorem invsym2 15378

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 2404	. . . . 5 |- ( Hom ` C ) = ( Hom ` C )
7	1, 2, 3, 4, 5, 6	invss 15376	. . . 4 |- ( ph -> ( Y N X ) C_ ( ( Y ( Hom ` C ) X ) X. ( X ( Hom ` C ) Y ) ) )
8		relxp 4933	. . . 4 |- Rel ( ( Y ( Hom ` C ) X ) X. ( X ( Hom ` C ) Y ) )
9		relss 4913	. . . 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 22	. . 3 |- ( ph -> Rel ( Y N X ) )
11		relcnv 5197	. . 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 15377	. . . 4 |- ( ph -> ( f ( X N Y ) g <-> g ( Y N X ) f ) )
14		vex 3064	. . . . . 6 |- g e. _V
15		vex 3064	. . . . . 6 |- f e. _V
16	14, 15	brcnv 5008	. . . . 5 |- ( g `' ( X N Y ) f <-> f ( X N Y ) g )
17		df-br 4398	. . . . 5 |- ( g `' ( X N Y ) f <-> <. g , f >. e. `' ( X N Y ) )
18	16, 17	bitr3i 253	. . . 4 |- ( f ( X N Y ) g <-> <. g , f >. e. `' ( X N Y ) )
19		df-br 4398	. . . 4 |- ( g ( Y N X ) f <-> <. g , f >. e. ( Y N X ) )
20	13, 18, 19	3bitr3g 289	. . 3 |- ( ph -> ( <. g , f >. e. `' ( X N Y ) <-> <. g , f >. e. ( Y N X ) ) )
21	20	eqrelrdv2 4925	. 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 1407   e. wcel 1844   C_ wss 3416   <.cop 3980   class class class wbr 4397   X. cxp 4823   `'ccnv 4824   Rel wrel 4830   ` cfv 5571  (class class class)co 6280   Basecbs 14843   Hom chom 14922   Catccat 15280  Invcinv 15360

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1641  ax-4 1654  ax-5 1727  ax-6 1773  ax-7 1816  ax-8 1846  ax-9 1848  ax-10 1863  ax-11 1868  ax-12 1880  ax-13 2028  ax-ext 2382  ax-rep 4509  ax-sep 4519  ax-nul 4527  ax-pow 4574  ax-pr 4632  ax-un 6576

This theorem depends on definitions:  df-bi 187  df-or 370  df-an 371  df-3an 978  df-tru 1410  df-ex 1636  df-nf 1640  df-sb 1766  df-eu 2244  df-mo 2245  df-clab 2390  df-cleq 2396  df-clel 2399  df-nfc 2554  df-ne 2602  df-ral 2761  df-rex 2762  df-reu 2763  df-rab 2765  df-v 3063  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3741  df-if 3888  df-pw 3959  df-sn 3975  df-pr 3977  df-op 3981  df-uni 4194  df-iun 4275  df-br 4398  df-opab 4456  df-mpt 4457  df-id 4740  df-xp 4831  df-rel 4832  df-cnv 4833  df-co 4834  df-dm 4835  df-rn 4836  df-res 4837  df-ima 4838  df-iota 5535  df-fun 5573  df-fn 5574  df-f 5575  df-f1 5576  df-fo 5577  df-f1o 5578  df-fv 5579  df-ov 6283  df-oprab 6284  df-mpt2 6285  df-1st 6786  df-2nd 6787  df-sect 15362  df-inv 15363

This theorem is referenced by:  invf  15383  invf1o  15384  invinv  15385  cicsym  15419