Theorem dfiso2 16432

Description: Alternate definition of an isomorphism of a category, according to definition 3.8 in [Adamek] p. 28. (Contributed by AV, 10-Apr-2017.)

Hypotheses

Ref	Expression
dfiso2.b	|- B = ( Base ` C )
dfiso2.h	|- H = ( Hom ` C )
dfiso2.c	|- ( ph -> C e. Cat )
dfiso2.i	|- I = ( Iso ` C )
dfiso2.x	|- ( ph -> X e. B )
dfiso2.y	|- ( ph -> Y e. B )
dfiso2.f	|- ( ph -> F e. ( X H Y ) )
dfiso2.1	|- .1. = ( Id ` C )
dfiso2.o	|- .o. = ( <. X , Y >. (comp ` C ) X )
dfiso2.p	|- .* = ( <. Y , X >. (comp ` C ) Y )

Assertion

Ref	Expression
dfiso2	|- ( ph -> ( F e. ( X I Y ) <-> E. g e. ( Y H X ) ( ( g .o. F ) = ( .1. ` X ) /\ ( F .* g ) = ( .1. ` Y ) ) ) )

Distinct variable groups:   C, g   g, F   g, H   g, I   g, X   g, Y   .o. , g   .* , g   .1. , g   ph, g

Allowed substitution hint:   B( g)

Proof of Theorem dfiso2

Dummy variable f is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfiso2.b	. . . 4 |- B = ( Base ` C )
2		eqid 2622	. . . 4 |- (Inv ` C ) = (Inv ` C )
3		dfiso2.c	. . . 4 |- ( ph -> C e. Cat )
4		dfiso2.x	. . . 4 |- ( ph -> X e. B )
5		dfiso2.y	. . . 4 |- ( ph -> Y e. B )
6		dfiso2.i	. . . 4 |- I = ( Iso ` C )
7	1, 2, 3, 4, 5, 6	isoval 16425	. . 3 |- ( ph -> ( X I Y ) = dom ( X (Inv ` C ) Y ) )
8	7	eleq2d 2687	. 2 |- ( ph -> ( F e. ( X I Y ) <-> F e. dom ( X (Inv ` C ) Y ) ) )
9		eqid 2622	. . . . 5 |- (Sect ` C ) = (Sect ` C )
10	1, 2, 3, 4, 5, 9	invfval 16419	. . . 4 |- ( ph -> ( X (Inv ` C ) Y ) = ( ( X (Sect ` C ) Y ) i^i `' ( Y (Sect ` C ) X ) ) )
11	10	dmeqd 5326	. . 3 |- ( ph -> dom ( X (Inv ` C ) Y ) = dom ( ( X (Sect ` C ) Y ) i^i `' ( Y (Sect ` C ) X ) ) )
12	11	eleq2d 2687	. 2 |- ( ph -> ( F e. dom ( X (Inv ` C ) Y ) <-> F e. dom ( ( X (Sect ` C ) Y ) i^i `' ( Y (Sect ` C ) X ) ) ) )
13		dfiso2.h	. . . . . . . . 9 |- H = ( Hom ` C )
14		eqid 2622	. . . . . . . . 9 |- (comp ` C ) = (comp ` C )
15		dfiso2.1	. . . . . . . . 9 |- .1. = ( Id ` C )
16	1, 13, 14, 15, 9, 3, 4, 5	sectfval 16411	. . . . . . . 8 |- ( ph -> ( X (Sect ` C ) Y ) = { <. f , g >. | ( ( f e. ( X H Y ) /\ g e. ( Y H X ) ) /\ ( g ( <. X , Y >. (comp ` C ) X ) f ) = ( .1. ` X ) ) } )
17	1, 13, 14, 15, 9, 3, 5, 4	sectfval 16411	. . . . . . . . . 10 |- ( ph -> ( Y (Sect ` C ) X ) = { <. g , f >. | ( ( g e. ( Y H X ) /\ f e. ( X H Y ) ) /\ ( f ( <. Y , X >. (comp ` C ) Y ) g ) = ( .1. ` Y ) ) } )
18	17	cnveqd 5298	. . . . . . . . 9 |- ( ph -> `' ( Y (Sect ` C ) X ) = `' { <. g , f >. | ( ( g e. ( Y H X ) /\ f e. ( X H Y ) ) /\ ( f ( <. Y , X >. (comp ` C ) Y ) g ) = ( .1. ` Y ) ) } )
19		cnvopab 5533	. . . . . . . . 9 |- `' { <. g , f >. | ( ( g e. ( Y H X ) /\ f e. ( X H Y ) ) /\ ( f ( <. Y , X >. (comp ` C ) Y ) g ) = ( .1. ` Y ) ) } = { <. f , g >. | ( ( g e. ( Y H X ) /\ f e. ( X H Y ) ) /\ ( f ( <. Y , X >. (comp ` C ) Y ) g ) = ( .1. ` Y ) ) }
20	18, 19	syl6eq 2672	. . . . . . . 8 |- ( ph -> `' ( Y (Sect ` C ) X ) = { <. f , g >. | ( ( g e. ( Y H X ) /\ f e. ( X H Y ) ) /\ ( f ( <. Y , X >. (comp ` C ) Y ) g ) = ( .1. ` Y ) ) } )
21	16, 20	ineq12d 3815	. . . . . . 7 |- ( ph -> ( ( X (Sect ` C ) Y ) i^i `' ( Y (Sect ` C ) X ) ) = ( { <. f , g >. | ( ( f e. ( X H Y ) /\ g e. ( Y H X ) ) /\ ( g ( <. X , Y >. (comp ` C ) X ) f ) = ( .1. ` X ) ) } i^i { <. f , g >. | ( ( g e. ( Y H X ) /\ f e. ( X H Y ) ) /\ ( f ( <. Y , X >. (comp ` C ) Y ) g ) = ( .1. ` Y ) ) } ) )
22		inopab 5252	. . . . . . . 8 |- ( { <. f , g >. | ( ( f e. ( X H Y ) /\ g e. ( Y H X ) ) /\ ( g ( <. X , Y >. (comp ` C ) X ) f ) = ( .1. ` X ) ) } i^i { <. f , g >. | ( ( g e. ( Y H X ) /\ f e. ( X H Y ) ) /\ ( f ( <. Y , X >. (comp ` C ) Y ) g ) = ( .1. ` Y ) ) } ) = { <. f , g >. | ( ( ( f e. ( X H Y ) /\ g e. ( Y H X ) ) /\ ( g ( <. X , Y >. (comp ` C ) X ) f ) = ( .1. ` X ) ) /\ ( ( g e. ( Y H X ) /\ f e. ( X H Y ) ) /\ ( f ( <. Y , X >. (comp ` C ) Y ) g ) = ( .1. ` Y ) ) ) }
23		an4 865	. . . . . . . . . 10 |- ( ( ( ( f e. ( X H Y ) /\ g e. ( Y H X ) ) /\ ( g ( <. X , Y >. (comp ` C ) X ) f ) = ( .1. ` X ) ) /\ ( ( g e. ( Y H X ) /\ f e. ( X H Y ) ) /\ ( f ( <. Y , X >. (comp ` C ) Y ) g ) = ( .1. ` Y ) ) ) <-> ( ( ( f e. ( X H Y ) /\ g e. ( Y H X ) ) /\ ( g e. ( Y H X ) /\ f e. ( X H Y ) ) ) /\ ( ( g ( <. X , Y >. (comp ` C ) X ) f ) = ( .1. ` X ) /\ ( f ( <. Y , X >. (comp ` C ) Y ) g ) = ( .1. ` Y ) ) ) )
24		an42 866	. . . . . . . . . . . 12 |- ( ( ( f e. ( X H Y ) /\ g e. ( Y H X ) ) /\ ( g e. ( Y H X ) /\ f e. ( X H Y ) ) ) <-> ( ( f e. ( X H Y ) /\ g e. ( Y H X ) ) /\ ( f e. ( X H Y ) /\ g e. ( Y H X ) ) ) )
25		anidm 676	. . . . . . . . . . . 12 |- ( ( ( f e. ( X H Y ) /\ g e. ( Y H X ) ) /\ ( f e. ( X H Y ) /\ g e. ( Y H X ) ) ) <-> ( f e. ( X H Y ) /\ g e. ( Y H X ) ) )
26	24, 25	bitri 264	. . . . . . . . . . 11 |- ( ( ( f e. ( X H Y ) /\ g e. ( Y H X ) ) /\ ( g e. ( Y H X ) /\ f e. ( X H Y ) ) ) <-> ( f e. ( X H Y ) /\ g e. ( Y H X ) ) )
27	26	anbi1i 731	. . . . . . . . . 10 |- ( ( ( ( f e. ( X H Y ) /\ g e. ( Y H X ) ) /\ ( g e. ( Y H X ) /\ f e. ( X H Y ) ) ) /\ ( ( g ( <. X , Y >. (comp ` C ) X ) f ) = ( .1. ` X ) /\ ( f ( <. Y , X >. (comp ` C ) Y ) g ) = ( .1. ` Y ) ) ) <-> ( ( f e. ( X H Y ) /\ g e. ( Y H X ) ) /\ ( ( g ( <. X , Y >. (comp ` C ) X ) f ) = ( .1. ` X ) /\ ( f ( <. Y , X >. (comp ` C ) Y ) g ) = ( .1. ` Y ) ) ) )
28	23, 27	bitri 264	. . . . . . . . 9 |- ( ( ( ( f e. ( X H Y ) /\ g e. ( Y H X ) ) /\ ( g ( <. X , Y >. (comp ` C ) X ) f ) = ( .1. ` X ) ) /\ ( ( g e. ( Y H X ) /\ f e. ( X H Y ) ) /\ ( f ( <. Y , X >. (comp ` C ) Y ) g ) = ( .1. ` Y ) ) ) <-> ( ( f e. ( X H Y ) /\ g e. ( Y H X ) ) /\ ( ( g ( <. X , Y >. (comp ` C ) X ) f ) = ( .1. ` X ) /\ ( f ( <. Y , X >. (comp ` C ) Y ) g ) = ( .1. ` Y ) ) ) )
29	28	opabbii 4717	. . . . . . . 8 |- { <. f , g >. | ( ( ( f e. ( X H Y ) /\ g e. ( Y H X ) ) /\ ( g ( <. X , Y >. (comp ` C ) X ) f ) = ( .1. ` X ) ) /\ ( ( g e. ( Y H X ) /\ f e. ( X H Y ) ) /\ ( f ( <. Y , X >. (comp ` C ) Y ) g ) = ( .1. ` Y ) ) ) } = { <. f , g >. | ( ( f e. ( X H Y ) /\ g e. ( Y H X ) ) /\ ( ( g ( <. X , Y >. (comp ` C ) X ) f ) = ( .1. ` X ) /\ ( f ( <. Y , X >. (comp ` C ) Y ) g ) = ( .1. ` Y ) ) ) }
30	22, 29	eqtri 2644	. . . . . . 7 |- ( { <. f , g >. | ( ( f e. ( X H Y ) /\ g e. ( Y H X ) ) /\ ( g ( <. X , Y >. (comp ` C ) X ) f ) = ( .1. ` X ) ) } i^i { <. f , g >. | ( ( g e. ( Y H X ) /\ f e. ( X H Y ) ) /\ ( f ( <. Y , X >. (comp ` C ) Y ) g ) = ( .1. ` Y ) ) } ) = { <. f , g >. | ( ( f e. ( X H Y ) /\ g e. ( Y H X ) ) /\ ( ( g ( <. X , Y >. (comp ` C ) X ) f ) = ( .1. ` X ) /\ ( f ( <. Y , X >. (comp ` C ) Y ) g ) = ( .1. ` Y ) ) ) }
31	21, 30	syl6eq 2672	. . . . . 6 |- ( ph -> ( ( X (Sect ` C ) Y ) i^i `' ( Y (Sect ` C ) X ) ) = { <. f , g >. | ( ( f e. ( X H Y ) /\ g e. ( Y H X ) ) /\ ( ( g ( <. X , Y >. (comp ` C ) X ) f ) = ( .1. ` X ) /\ ( f ( <. Y , X >. (comp ` C ) Y ) g ) = ( .1. ` Y ) ) ) } )
32	31	dmeqd 5326	. . . . 5 |- ( ph -> dom ( ( X (Sect ` C ) Y ) i^i `' ( Y (Sect ` C ) X ) ) = dom { <. f , g >. | ( ( f e. ( X H Y ) /\ g e. ( Y H X ) ) /\ ( ( g ( <. X , Y >. (comp ` C ) X ) f ) = ( .1. ` X ) /\ ( f ( <. Y , X >. (comp ` C ) Y ) g ) = ( .1. ` Y ) ) ) } )
33		dmopab 5335	. . . . 5 |- dom { <. f , g >. | ( ( f e. ( X H Y ) /\ g e. ( Y H X ) ) /\ ( ( g ( <. X , Y >. (comp ` C ) X ) f ) = ( .1. ` X ) /\ ( f ( <. Y , X >. (comp ` C ) Y ) g ) = ( .1. ` Y ) ) ) } = { f | E. g ( ( f e. ( X H Y ) /\ g e. ( Y H X ) ) /\ ( ( g ( <. X , Y >. (comp ` C ) X ) f ) = ( .1. ` X ) /\ ( f ( <. Y , X >. (comp ` C ) Y ) g ) = ( .1. ` Y ) ) ) }
34	32, 33	syl6eq 2672	. . . 4 |- ( ph -> dom ( ( X (Sect ` C ) Y ) i^i `' ( Y (Sect ` C ) X ) ) = { f | E. g ( ( f e. ( X H Y ) /\ g e. ( Y H X ) ) /\ ( ( g ( <. X , Y >. (comp ` C ) X ) f ) = ( .1. ` X ) /\ ( f ( <. Y , X >. (comp ` C ) Y ) g ) = ( .1. ` Y ) ) ) } )
35	34	eleq2d 2687	. . 3 |- ( ph -> ( F e. dom ( ( X (Sect ` C ) Y ) i^i `' ( Y (Sect ` C ) X ) ) <-> F e. { f | E. g ( ( f e. ( X H Y ) /\ g e. ( Y H X ) ) /\ ( ( g ( <. X , Y >. (comp ` C ) X ) f ) = ( .1. ` X ) /\ ( f ( <. Y , X >. (comp ` C ) Y ) g ) = ( .1. ` Y ) ) ) } ) )
36		dfiso2.f	. . . 4 |- ( ph -> F e. ( X H Y ) )
37		eleq1 2689	. . . . . . . 8 |- ( f = F -> ( f e. ( X H Y ) <-> F e. ( X H Y ) ) )
38	37	anbi1d 741	. . . . . . 7 |- ( f = F -> ( ( f e. ( X H Y ) /\ g e. ( Y H X ) ) <-> ( F e. ( X H Y ) /\ g e. ( Y H X ) ) ) )
39		oveq2 6658	. . . . . . . . 9 |- ( f = F -> ( g ( <. X , Y >. (comp ` C ) X ) f ) = ( g ( <. X , Y >. (comp ` C ) X ) F ) )
40	39	eqeq1d 2624	. . . . . . . 8 |- ( f = F -> ( ( g ( <. X , Y >. (comp ` C ) X ) f ) = ( .1. ` X ) <-> ( g ( <. X , Y >. (comp ` C ) X ) F ) = ( .1. ` X ) ) )
41		oveq1 6657	. . . . . . . . 9 |- ( f = F -> ( f ( <. Y , X >. (comp ` C ) Y ) g ) = ( F ( <. Y , X >. (comp ` C ) Y ) g ) )
42	41	eqeq1d 2624	. . . . . . . 8 |- ( f = F -> ( ( f ( <. Y , X >. (comp ` C ) Y ) g ) = ( .1. ` Y ) <-> ( F ( <. Y , X >. (comp ` C ) Y ) g ) = ( .1. ` Y ) ) )
43	40, 42	anbi12d 747	. . . . . . 7 |- ( f = F -> ( ( ( g ( <. X , Y >. (comp ` C ) X ) f ) = ( .1. ` X ) /\ ( f ( <. Y , X >. (comp ` C ) Y ) g ) = ( .1. ` Y ) ) <-> ( ( g ( <. X , Y >. (comp ` C ) X ) F ) = ( .1. ` X ) /\ ( F ( <. Y , X >. (comp ` C ) Y ) g ) = ( .1. ` Y ) ) ) )
44	38, 43	anbi12d 747	. . . . . 6 |- ( f = F -> ( ( ( f e. ( X H Y ) /\ g e. ( Y H X ) ) /\ ( ( g ( <. X , Y >. (comp ` C ) X ) f ) = ( .1. ` X ) /\ ( f ( <. Y , X >. (comp ` C ) Y ) g ) = ( .1. ` Y ) ) ) <-> ( ( F e. ( X H Y ) /\ g e. ( Y H X ) ) /\ ( ( g ( <. X , Y >. (comp ` C ) X ) F ) = ( .1. ` X ) /\ ( F ( <. Y , X >. (comp ` C ) Y ) g ) = ( .1. ` Y ) ) ) ) )
45	44	exbidv 1850	. . . . 5 |- ( f = F -> ( E. g ( ( f e. ( X H Y ) /\ g e. ( Y H X ) ) /\ ( ( g ( <. X , Y >. (comp ` C ) X ) f ) = ( .1. ` X ) /\ ( f ( <. Y , X >. (comp ` C ) Y ) g ) = ( .1. ` Y ) ) ) <-> E. g ( ( F e. ( X H Y ) /\ g e. ( Y H X ) ) /\ ( ( g ( <. X , Y >. (comp ` C ) X ) F ) = ( .1. ` X ) /\ ( F ( <. Y , X >. (comp ` C ) Y ) g ) = ( .1. ` Y ) ) ) ) )
46	45	elabg 3351	. . . 4 |- ( F e. ( X H Y ) -> ( F e. { f | E. g ( ( f e. ( X H Y ) /\ g e. ( Y H X ) ) /\ ( ( g ( <. X , Y >. (comp ` C ) X ) f ) = ( .1. ` X ) /\ ( f ( <. Y , X >. (comp ` C ) Y ) g ) = ( .1. ` Y ) ) ) } <-> E. g ( ( F e. ( X H Y ) /\ g e. ( Y H X ) ) /\ ( ( g ( <. X , Y >. (comp ` C ) X ) F ) = ( .1. ` X ) /\ ( F ( <. Y , X >. (comp ` C ) Y ) g ) = ( .1. ` Y ) ) ) ) )
47	36, 46	syl 17	. . 3 |- ( ph -> ( F e. { f | E. g ( ( f e. ( X H Y ) /\ g e. ( Y H X ) ) /\ ( ( g ( <. X , Y >. (comp ` C ) X ) f ) = ( .1. ` X ) /\ ( f ( <. Y , X >. (comp ` C ) Y ) g ) = ( .1. ` Y ) ) ) } <-> E. g ( ( F e. ( X H Y ) /\ g e. ( Y H X ) ) /\ ( ( g ( <. X , Y >. (comp ` C ) X ) F ) = ( .1. ` X ) /\ ( F ( <. Y , X >. (comp ` C ) Y ) g ) = ( .1. ` Y ) ) ) ) )
48	36	biantrurd 529	. . . . . . 7 |- ( ph -> ( g e. ( Y H X ) <-> ( F e. ( X H Y ) /\ g e. ( Y H X ) ) ) )
49	48	bicomd 213	. . . . . 6 |- ( ph -> ( ( F e. ( X H Y ) /\ g e. ( Y H X ) ) <-> g e. ( Y H X ) ) )
50		dfiso2.o	. . . . . . . . . . 11 |- .o. = ( <. X , Y >. (comp ` C ) X )
51	50	a1i 11	. . . . . . . . . 10 |- ( ph -> .o. = ( <. X , Y >. (comp ` C ) X ) )
52	51	eqcomd 2628	. . . . . . . . 9 |- ( ph -> ( <. X , Y >. (comp ` C ) X ) = .o. )
53	52	oveqd 6667	. . . . . . . 8 |- ( ph -> ( g ( <. X , Y >. (comp ` C ) X ) F ) = ( g .o. F ) )
54	53	eqeq1d 2624	. . . . . . 7 |- ( ph -> ( ( g ( <. X , Y >. (comp ` C ) X ) F ) = ( .1. ` X ) <-> ( g .o. F ) = ( .1. ` X ) ) )
55		dfiso2.p	. . . . . . . . . . 11 |- .* = ( <. Y , X >. (comp ` C ) Y )
56	55	a1i 11	. . . . . . . . . 10 |- ( ph -> .* = ( <. Y , X >. (comp ` C ) Y ) )
57	56	eqcomd 2628	. . . . . . . . 9 |- ( ph -> ( <. Y , X >. (comp ` C ) Y ) = .* )
58	57	oveqd 6667	. . . . . . . 8 |- ( ph -> ( F ( <. Y , X >. (comp ` C ) Y ) g ) = ( F .* g ) )
59	58	eqeq1d 2624	. . . . . . 7 |- ( ph -> ( ( F ( <. Y , X >. (comp ` C ) Y ) g ) = ( .1. ` Y ) <-> ( F .* g ) = ( .1. ` Y ) ) )
60	54, 59	anbi12d 747	. . . . . 6 |- ( ph -> ( ( ( g ( <. X , Y >. (comp ` C ) X ) F ) = ( .1. ` X ) /\ ( F ( <. Y , X >. (comp ` C ) Y ) g ) = ( .1. ` Y ) ) <-> ( ( g .o. F ) = ( .1. ` X ) /\ ( F .* g ) = ( .1. ` Y ) ) ) )
61	49, 60	anbi12d 747	. . . . 5 |- ( ph -> ( ( ( F e. ( X H Y ) /\ g e. ( Y H X ) ) /\ ( ( g ( <. X , Y >. (comp ` C ) X ) F ) = ( .1. ` X ) /\ ( F ( <. Y , X >. (comp ` C ) Y ) g ) = ( .1. ` Y ) ) ) <-> ( g e. ( Y H X ) /\ ( ( g .o. F ) = ( .1. ` X ) /\ ( F .* g ) = ( .1. ` Y ) ) ) ) )
62	61	exbidv 1850	. . . 4 |- ( ph -> ( E. g ( ( F e. ( X H Y ) /\ g e. ( Y H X ) ) /\ ( ( g ( <. X , Y >. (comp ` C ) X ) F ) = ( .1. ` X ) /\ ( F ( <. Y , X >. (comp ` C ) Y ) g ) = ( .1. ` Y ) ) ) <-> E. g ( g e. ( Y H X ) /\ ( ( g .o. F ) = ( .1. ` X ) /\ ( F .* g ) = ( .1. ` Y ) ) ) ) )
63		df-rex 2918	. . . 4 |- ( E. g e. ( Y H X ) ( ( g .o. F ) = ( .1. ` X ) /\ ( F .* g ) = ( .1. ` Y ) ) <-> E. g ( g e. ( Y H X ) /\ ( ( g .o. F ) = ( .1. ` X ) /\ ( F .* g ) = ( .1. ` Y ) ) ) )
64	62, 63	syl6bbr 278	. . 3 |- ( ph -> ( E. g ( ( F e. ( X H Y ) /\ g e. ( Y H X ) ) /\ ( ( g ( <. X , Y >. (comp ` C ) X ) F ) = ( .1. ` X ) /\ ( F ( <. Y , X >. (comp ` C ) Y ) g ) = ( .1. ` Y ) ) ) <-> E. g e. ( Y H X ) ( ( g .o. F ) = ( .1. ` X ) /\ ( F .* g ) = ( .1. ` Y ) ) ) )
65	35, 47, 64	3bitrd 294	. 2 |- ( ph -> ( F e. dom ( ( X (Sect ` C ) Y ) i^i `' ( Y (Sect ` C ) X ) ) <-> E. g e. ( Y H X ) ( ( g .o. F ) = ( .1. ` X ) /\ ( F .* g ) = ( .1. ` Y ) ) ) )
66	8, 12, 65	3bitrd 294	1 |- ( ph -> ( F e. ( X I Y ) <-> E. g e. ( Y H X ) ( ( g .o. F ) = ( .1. ` X ) /\ ( F .* g ) = ( .1. ` Y ) ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   E.wex 1704   e. wcel 1990   {cab 2608   E.wrex 2913   i^i cin 3573   <.cop 4183   {copab 4712   `'ccnv 5113   dom cdm 5114   ` 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-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:  dfiso3  16433