Theorem invisoinvl 16450

Description: The inverse of an isomorphism F (which is unique because of invf 16428 and is therefore denoted by ( ( X N Y ) ` F ), see also remark 3.12 in [Adamek] p. 28) is invers to the isomorphism. (Contributed by AV, 9-Apr-2017.)

Hypotheses

Ref	Expression
invisoinv.b	|- B = ( Base ` C )
invisoinv.i	|- I = ( Iso ` C )
invisoinv.n	|- N = (Inv ` C )
invisoinv.c	|- ( ph -> C e. Cat )
invisoinv.x	|- ( ph -> X e. B )
invisoinv.y	|- ( ph -> Y e. B )
invisoinv.f	|- ( ph -> F e. ( X I Y ) )

Assertion

Ref	Expression
invisoinvl	|- ( ph -> ( ( X N Y ) ` F ) ( Y N X ) F )

Proof of Theorem invisoinvl

Step	Hyp	Ref	Expression
1		invisoinv.b	. . . 4 |- B = ( Base ` C )
2		invisoinv.n	. . . 4 |- N = (Inv ` C )
3		invisoinv.c	. . . 4 |- ( ph -> C e. Cat )
4		invisoinv.x	. . . 4 |- ( ph -> X e. B )
5		invisoinv.y	. . . 4 |- ( ph -> Y e. B )
6		invisoinv.i	. . . 4 |- I = ( Iso ` C )
7		invisoinv.f	. . . 4 |- ( ph -> F e. ( X I Y ) )
8		eqid 2622	. . . 4 |- (comp ` C ) = (comp ` C )
9		eqid 2622	. . . . . 6 |- ( Id ` C ) = ( Id ` C )
10	1, 9, 3, 5	idiso 16448	. . . . 5 |- ( ph -> ( ( Id ` C ) ` Y ) e. ( Y ( Iso ` C ) Y ) )
11	6	a1i 11	. . . . . 6 |- ( ph -> I = ( Iso ` C ) )
12	11	oveqd 6667	. . . . 5 |- ( ph -> ( Y I Y ) = ( Y ( Iso ` C ) Y ) )
13	10, 12	eleqtrrd 2704	. . . 4 |- ( ph -> ( ( Id ` C ) ` Y ) e. ( Y I Y ) )
14	1, 2, 3, 4, 5, 6, 7, 8, 5, 13	invco 16431	. . 3 |- ( ph -> ( ( ( Id ` C ) ` Y ) ( <. X , Y >. (comp ` C ) Y ) F ) ( X N Y ) ( ( ( X N Y ) ` F ) ( <. Y , Y >. (comp ` C ) X ) ( ( Y N Y ) ` ( ( Id ` C ) ` Y ) ) ) )
15		eqid 2622	. . . 4 |- ( Hom ` C ) = ( Hom ` C )
16	1, 15, 6, 3, 4, 5	isohom 16436	. . . . 5 |- ( ph -> ( X I Y ) C_ ( X ( Hom ` C ) Y ) )
17	16, 7	sseldd 3604	. . . 4 |- ( ph -> F e. ( X ( Hom ` C ) Y ) )
18	1, 15, 9, 3, 4, 8, 5, 17	catlid 16344	. . 3 |- ( ph -> ( ( ( Id ` C ) ` Y ) ( <. X , Y >. (comp ` C ) Y ) F ) = F )
19	2	a1i 11	. . . . . . . 8 |- ( ph -> N = (Inv ` C ) )
20	19	oveqd 6667	. . . . . . 7 |- ( ph -> ( Y N Y ) = ( Y (Inv ` C ) Y ) )
21	20	fveq1d 6193	. . . . . 6 |- ( ph -> ( ( Y N Y ) ` ( ( Id ` C ) ` Y ) ) = ( ( Y (Inv ` C ) Y ) ` ( ( Id ` C ) ` Y ) ) )
22	1, 9, 3, 5	idinv 16449	. . . . . 6 |- ( ph -> ( ( Y (Inv ` C ) Y ) ` ( ( Id ` C ) ` Y ) ) = ( ( Id ` C ) ` Y ) )
23	21, 22	eqtrd 2656	. . . . 5 |- ( ph -> ( ( Y N Y ) ` ( ( Id ` C ) ` Y ) ) = ( ( Id ` C ) ` Y ) )
24	23	oveq2d 6666	. . . 4 |- ( ph -> ( ( ( X N Y ) ` F ) ( <. Y , Y >. (comp ` C ) X ) ( ( Y N Y ) ` ( ( Id ` C ) ` Y ) ) ) = ( ( ( X N Y ) ` F ) ( <. Y , Y >. (comp ` C ) X ) ( ( Id ` C ) ` Y ) ) )
25	1, 15, 6, 3, 5, 4	isohom 16436	. . . . . 6 |- ( ph -> ( Y I X ) C_ ( Y ( Hom ` C ) X ) )
26	1, 2, 3, 4, 5, 6	invf 16428	. . . . . . 7 |- ( ph -> ( X N Y ) : ( X I Y ) --> ( Y I X ) )
27	26, 7	ffvelrnd 6360	. . . . . 6 |- ( ph -> ( ( X N Y ) ` F ) e. ( Y I X ) )
28	25, 27	sseldd 3604	. . . . 5 |- ( ph -> ( ( X N Y ) ` F ) e. ( Y ( Hom ` C ) X ) )
29	1, 15, 9, 3, 5, 8, 4, 28	catrid 16345	. . . 4 |- ( ph -> ( ( ( X N Y ) ` F ) ( <. Y , Y >. (comp ` C ) X ) ( ( Id ` C ) ` Y ) ) = ( ( X N Y ) ` F ) )
30	24, 29	eqtrd 2656	. . 3 |- ( ph -> ( ( ( X N Y ) ` F ) ( <. Y , Y >. (comp ` C ) X ) ( ( Y N Y ) ` ( ( Id ` C ) ` Y ) ) ) = ( ( X N Y ) ` F ) )
31	14, 18, 30	3brtr3d 4684	. 2 |- ( ph -> F ( X N Y ) ( ( X N Y ) ` F ) )
32	1, 2, 3, 5, 4	invsym 16422	. 2 |- ( ph -> ( ( ( X N Y ) ` F ) ( Y N X ) F <-> F ( X N Y ) ( ( X N Y ) ` F ) ) )
33	31, 32	mpbird 247	1 |- ( ph -> ( ( X N Y ) ` F ) ( Y N X ) F )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   <.cop 4183   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   Basecbs 15857   Hom chom 15952  compcco 15953   Catccat 16325   Idccid 16326  Invcinv 16405   Iso ciso 16406

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-cat 16329  df-cid 16330  df-sect 16407  df-inv 16408  df-iso 16409

This theorem is referenced by:  invisoinvr  16451  isocoinvid  16453