Theorem dfiso3 16433

Description: Alternate definition of an isomorphism of a category as a section in both directions. (Contributed by AV, 11-Apr-2017.)

Hypotheses

Ref	Expression
dfiso3.b	|- B = ( Base ` C )
dfiso3.h	|- H = ( Hom ` C )
dfiso3.i	|- I = ( Iso ` C )
dfiso3.s	|- S = (Sect ` C )
dfiso3.c	|- ( ph -> C e. Cat )
dfiso3.x	|- ( ph -> X e. B )
dfiso3.y	|- ( ph -> Y e. B )
dfiso3.f	|- ( ph -> F e. ( X H Y ) )

Assertion

Ref	Expression
dfiso3	|- ( ph -> ( F e. ( X I Y ) <-> E. g e. ( Y H X ) ( g ( Y S X ) F /\ F ( X S Y ) g ) ) )

Distinct variable groups:   C, g   g, F   g, H   g, I   g, X   g, Y   ph, g

Allowed substitution hints:   B( g)   S( g)

Proof of Theorem dfiso3

Step	Hyp	Ref	Expression
1		dfiso3.b	. . 3 |- B = ( Base ` C )
2		dfiso3.h	. . 3 |- H = ( Hom ` C )
3		dfiso3.c	. . 3 |- ( ph -> C e. Cat )
4		dfiso3.i	. . 3 |- I = ( Iso ` C )
5		dfiso3.x	. . 3 |- ( ph -> X e. B )
6		dfiso3.y	. . 3 |- ( ph -> Y e. B )
7		dfiso3.f	. . 3 |- ( ph -> F e. ( X H Y ) )
8		eqid 2622	. . 3 |- ( Id ` C ) = ( Id ` C )
9		eqid 2622	. . 3 |- ( <. X , Y >. (comp ` C ) X ) = ( <. X , Y >. (comp ` C ) X )
10		eqid 2622	. . 3 |- ( <. Y , X >. (comp ` C ) Y ) = ( <. Y , X >. (comp ` C ) Y )
11	1, 2, 3, 4, 5, 6, 7, 8, 9, 10	dfiso2 16432	. 2 |- ( ph -> ( F e. ( X I Y ) <-> E. g e. ( Y H X ) ( ( g ( <. X , Y >. (comp ` C ) X ) F ) = ( ( Id ` C ) ` X ) /\ ( F ( <. Y , X >. (comp ` C ) Y ) g ) = ( ( Id ` C ) ` Y ) ) ) )
12		eqid 2622	. . . . . 6 |- (comp ` C ) = (comp ` C )
13		dfiso3.s	. . . . . 6 |- S = (Sect ` C )
14	3	adantr 481	. . . . . 6 |- ( ( ph /\ g e. ( Y H X ) ) -> C e. Cat )
15	6	adantr 481	. . . . . 6 |- ( ( ph /\ g e. ( Y H X ) ) -> Y e. B )
16	5	adantr 481	. . . . . 6 |- ( ( ph /\ g e. ( Y H X ) ) -> X e. B )
17		simpr 477	. . . . . 6 |- ( ( ph /\ g e. ( Y H X ) ) -> g e. ( Y H X ) )
18	7	adantr 481	. . . . . 6 |- ( ( ph /\ g e. ( Y H X ) ) -> F e. ( X H Y ) )
19	1, 2, 12, 8, 13, 14, 15, 16, 17, 18	issect2 16414	. . . . 5 |- ( ( ph /\ g e. ( Y H X ) ) -> ( g ( Y S X ) F <-> ( F ( <. Y , X >. (comp ` C ) Y ) g ) = ( ( Id ` C ) ` Y ) ) )
20	1, 2, 12, 8, 13, 14, 16, 15, 18, 17	issect2 16414	. . . . 5 |- ( ( ph /\ g e. ( Y H X ) ) -> ( F ( X S Y ) g <-> ( g ( <. X , Y >. (comp ` C ) X ) F ) = ( ( Id ` C ) ` X ) ) )
21	19, 20	anbi12d 747	. . . 4 |- ( ( ph /\ g e. ( Y H X ) ) -> ( ( g ( Y S X ) F /\ F ( X S Y ) g ) <-> ( ( F ( <. Y , X >. (comp ` C ) Y ) g ) = ( ( Id ` C ) ` Y ) /\ ( g ( <. X , Y >. (comp ` C ) X ) F ) = ( ( Id ` C ) ` X ) ) ) )
22		ancom 466	. . . 4 |- ( ( ( F ( <. Y , X >. (comp ` C ) Y ) g ) = ( ( Id ` C ) ` Y ) /\ ( g ( <. X , Y >. (comp ` C ) X ) F ) = ( ( Id ` C ) ` X ) ) <-> ( ( g ( <. X , Y >. (comp ` C ) X ) F ) = ( ( Id ` C ) ` X ) /\ ( F ( <. Y , X >. (comp ` C ) Y ) g ) = ( ( Id ` C ) ` Y ) ) )
23	21, 22	syl6rbb 277	. . 3 |- ( ( ph /\ g e. ( Y H X ) ) -> ( ( ( g ( <. X , Y >. (comp ` C ) X ) F ) = ( ( Id ` C ) ` X ) /\ ( F ( <. Y , X >. (comp ` C ) Y ) g ) = ( ( Id ` C ) ` Y ) ) <-> ( g ( Y S X ) F /\ F ( X S Y ) g ) ) )
24	23	rexbidva 3049	. 2 |- ( ph -> ( E. g e. ( Y H X ) ( ( g ( <. X , Y >. (comp ` C ) X ) F ) = ( ( Id ` C ) ` X ) /\ ( F ( <. Y , X >. (comp ` C ) Y ) g ) = ( ( Id ` C ) ` Y ) ) <-> E. g e. ( Y H X ) ( g ( Y S X ) F /\ F ( X S Y ) g ) ) )
25	11, 24	bitrd 268	1 |- ( ph -> ( F e. ( X I Y ) <-> E. g e. ( Y H X ) ( g ( Y S X ) F /\ F ( X S Y ) g ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   E.wrex 2913   <.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   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-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-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-sect 16407  df-inv 16408  df-iso 16409

This theorem is referenced by: (None)