Theorem isocoinvid 16453

Description: The inverse of an isomorphism composed with the isomorphism is the identity. (Contributed by AV, 10-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 ) )
invcoisoid.1	|- .1. = ( Id ` C )
isocoinvid.o	|- .o. = ( <. Y , X >. (comp ` C ) Y )

Assertion

Ref	Expression
isocoinvid	|- ( ph -> ( F .o. ( ( X N Y ) ` F ) ) = ( .1. ` Y ) )

Proof of Theorem isocoinvid

Step	Hyp	Ref	Expression
1		invisoinv.b	. . . 4 |- B = ( Base ` C )
2		invisoinv.i	. . . 4 |- I = ( Iso ` C )
3		invisoinv.n	. . . 4 |- N = (Inv ` C )
4		invisoinv.c	. . . 4 |- ( ph -> C e. Cat )
5		invisoinv.x	. . . 4 |- ( ph -> X e. B )
6		invisoinv.y	. . . 4 |- ( ph -> Y e. B )
7		invisoinv.f	. . . 4 |- ( ph -> F e. ( X I Y ) )
8	1, 2, 3, 4, 5, 6, 7	invisoinvl 16450	. . 3 |- ( ph -> ( ( X N Y ) ` F ) ( Y N X ) F )
9		eqid 2622	. . . . 5 |- (Sect ` C ) = (Sect ` C )
10	1, 3, 4, 6, 5, 9	isinv 16420	. . . 4 |- ( ph -> ( ( ( X N Y ) ` F ) ( Y N X ) F <-> ( ( ( X N Y ) ` F ) ( Y (Sect ` C ) X ) F /\ F ( X (Sect ` C ) Y ) ( ( X N Y ) ` F ) ) ) )
11		simpl 473	. . . 4 |- ( ( ( ( X N Y ) ` F ) ( Y (Sect ` C ) X ) F /\ F ( X (Sect ` C ) Y ) ( ( X N Y ) ` F ) ) -> ( ( X N Y ) ` F ) ( Y (Sect ` C ) X ) F )
12	10, 11	syl6bi 243	. . 3 |- ( ph -> ( ( ( X N Y ) ` F ) ( Y N X ) F -> ( ( X N Y ) ` F ) ( Y (Sect ` C ) X ) F ) )
13	8, 12	mpd 15	. 2 |- ( ph -> ( ( X N Y ) ` F ) ( Y (Sect ` C ) X ) F )
14		eqid 2622	. . . 4 |- ( Hom ` C ) = ( Hom ` C )
15		eqid 2622	. . . 4 |- (comp ` C ) = (comp ` C )
16		invcoisoid.1	. . . 4 |- .1. = ( Id ` C )
17	1, 14, 2, 4, 6, 5	isohom 16436	. . . . 5 |- ( ph -> ( Y I X ) C_ ( Y ( Hom ` C ) X ) )
18	1, 3, 4, 5, 6, 2	invf 16428	. . . . . 6 |- ( ph -> ( X N Y ) : ( X I Y ) --> ( Y I X ) )
19	18, 7	ffvelrnd 6360	. . . . 5 |- ( ph -> ( ( X N Y ) ` F ) e. ( Y I X ) )
20	17, 19	sseldd 3604	. . . 4 |- ( ph -> ( ( X N Y ) ` F ) e. ( Y ( Hom ` C ) X ) )
21	1, 14, 2, 4, 5, 6	isohom 16436	. . . . 5 |- ( ph -> ( X I Y ) C_ ( X ( Hom ` C ) Y ) )
22	21, 7	sseldd 3604	. . . 4 |- ( ph -> F e. ( X ( Hom ` C ) Y ) )
23	1, 14, 15, 16, 9, 4, 6, 5, 20, 22	issect2 16414	. . 3 |- ( ph -> ( ( ( X N Y ) ` F ) ( Y (Sect ` C ) X ) F <-> ( F ( <. Y , X >. (comp ` C ) Y ) ( ( X N Y ) ` F ) ) = ( .1. ` Y ) ) )
24		isocoinvid.o	. . . . . . 7 |- .o. = ( <. Y , X >. (comp ` C ) Y )
25	24	a1i 11	. . . . . 6 |- ( ph -> .o. = ( <. Y , X >. (comp ` C ) Y ) )
26	25	eqcomd 2628	. . . . 5 |- ( ph -> ( <. Y , X >. (comp ` C ) Y ) = .o. )
27	26	oveqd 6667	. . . 4 |- ( ph -> ( F ( <. Y , X >. (comp ` C ) Y ) ( ( X N Y ) ` F ) ) = ( F .o. ( ( X N Y ) ` F ) ) )
28	27	eqeq1d 2624	. . 3 |- ( ph -> ( ( F ( <. Y , X >. (comp ` C ) Y ) ( ( X N Y ) ` F ) ) = ( .1. ` Y ) <-> ( F .o. ( ( X N Y ) ` F ) ) = ( .1. ` Y ) ) )
29	23, 28	bitrd 268	. 2 |- ( ph -> ( ( ( X N Y ) ` F ) ( Y (Sect ` C ) X ) F <-> ( F .o. ( ( X N Y ) ` F ) ) = ( .1. ` Y ) ) )
30	13, 29	mpbid 222	1 |- ( ph -> ( F .o. ( ( X N Y ) ` F ) ) = ( .1. ` Y ) )

Syntax hints:   -> wi 4   /\ wa 384   = 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  Sectcsect 16404  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: (None)