Theorem invsym2 15661

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 2423	. . . . 5 |- ( Hom ` C ) = ( Hom ` C )
7	1, 2, 3, 4, 5, 6	invss 15659	. . . 4 |- ( ph -> ( Y N X ) C_ ( ( Y ( Hom ` C ) X ) X. ( X ( Hom ` C ) Y ) ) )
8		relxp 4959	. . . 4 |- Rel ( ( Y ( Hom ` C ) X ) X. ( X ( Hom ` C ) Y ) )
9		relss 4939	. . . 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 5224	. . 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 15660	. . . 4 |- ( ph -> ( f ( X N Y ) g <-> g ( Y N X ) f ) )
14		vex 3085	. . . . . 6 |- g e. _V
15		vex 3085	. . . . . 6 |- f e. _V
16	14, 15	brcnv 5034	. . . . 5 |- ( g `' ( X N Y ) f <-> f ( X N Y ) g )
17		df-br 4422	. . . . 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 4422	. . . 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 4951	. 2 |- ( ( ( Rel `' ( X N Y ) /\ Rel ( Y N X ) ) /\ ph ) -> `' ( X N Y ) = ( Y N X ) )
22	12, 21	mpancom 674	1 |- ( ph -> `' ( X N Y ) = ( Y N X ) )

Syntax hints:   -> wi 4   /\ wa 371   = wceq 1438   e. wcel 1869   C_ wss 3437   <.cop 4003   class class class wbr 4421   X. cxp 4849   `'ccnv 4850   Rel wrel 4856   ` cfv 5599  (class class class)co 6303   Basecbs 15114   Hom chom 15194   Catccat 15563  Invcinv 15643

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1666  ax-4 1679  ax-5 1749  ax-6 1795  ax-7 1840  ax-8 1871  ax-9 1873  ax-10 1888  ax-11 1893  ax-12 1906  ax-13 2054  ax-ext 2401  ax-rep 4534  ax-sep 4544  ax-nul 4553  ax-pow 4600  ax-pr 4658  ax-un 6595

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 985  df-tru 1441  df-ex 1661  df-nf 1665  df-sb 1788  df-eu 2270  df-mo 2271  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2573  df-ne 2621  df-ral 2781  df-rex 2782  df-reu 2783  df-rab 2785  df-v 3084  df-sbc 3301  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3763  df-if 3911  df-pw 3982  df-sn 3998  df-pr 4000  df-op 4004  df-uni 4218  df-iun 4299  df-br 4422  df-opab 4481  df-mpt 4482  df-id 4766  df-xp 4857  df-rel 4858  df-cnv 4859  df-co 4860  df-dm 4861  df-rn 4862  df-res 4863  df-ima 4864  df-iota 5563  df-fun 5601  df-fn 5602  df-f 5603  df-f1 5604  df-fo 5605  df-f1o 5606  df-fv 5607  df-ov 6306  df-oprab 6307  df-mpt2 6308  df-1st 6805  df-2nd 6806  df-sect 15645  df-inv 15646

This theorem is referenced by:  invf  15666  invf1o  15667  invinv  15668  cicsym  15702