Theorem fuciso 16635

Description: A natural transformation is an isomorphism of functors iff all its components are isomorphisms. (Contributed by Mario Carneiro, 28-Jan-2017.)

Hypotheses

Ref	Expression
fuciso.q	|- Q = ( C FuncCat D )
fuciso.b	|- B = ( Base ` C )
fuciso.n	|- N = ( C Nat D )
fuciso.f	|- ( ph -> F e. ( C Func D ) )
fuciso.g	|- ( ph -> G e. ( C Func D ) )
fuciso.i	|- I = ( Iso ` Q )
fuciso.j	|- J = ( Iso ` D )

Assertion

Ref	Expression
fuciso	|- ( ph -> ( A e. ( F I G ) <-> ( A e. ( F N G ) /\ A. x e. B ( A ` x ) e. ( ( ( 1st ` F ) ` x ) J ( ( 1st ` G ) ` x ) ) ) ) )

Distinct variable groups:   x, A   x, B   x, C   x, D   x, I   x, F   x, G   x, J   x, N   ph, x   x, Q

Proof of Theorem fuciso

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fuciso.q	. . . . . 6 |- Q = ( C FuncCat D )
2	1	fucbas 16620	. . . . 5 |- ( C Func D ) = ( Base ` Q )
3		fuciso.n	. . . . . 6 |- N = ( C Nat D )
4	1, 3	fuchom 16621	. . . . 5 |- N = ( Hom ` Q )
5		fuciso.i	. . . . 5 |- I = ( Iso ` Q )
6		fuciso.f	. . . . . . . 8 |- ( ph -> F e. ( C Func D ) )
7		funcrcl 16523	. . . . . . . 8 |- ( F e. ( C Func D ) -> ( C e. Cat /\ D e. Cat ) )
8	6, 7	syl 17	. . . . . . 7 |- ( ph -> ( C e. Cat /\ D e. Cat ) )
9	8	simpld 475	. . . . . 6 |- ( ph -> C e. Cat )
10	8	simprd 479	. . . . . 6 |- ( ph -> D e. Cat )
11	1, 9, 10	fuccat 16630	. . . . 5 |- ( ph -> Q e. Cat )
12		fuciso.g	. . . . 5 |- ( ph -> G e. ( C Func D ) )
13	2, 4, 5, 11, 6, 12	isohom 16436	. . . 4 |- ( ph -> ( F I G ) C_ ( F N G ) )
14	13	sselda 3603	. . 3 |- ( ( ph /\ A e. ( F I G ) ) -> A e. ( F N G ) )
15		eqid 2622	. . . . 5 |- ( Base ` D ) = ( Base ` D )
16		eqid 2622	. . . . 5 |- (Inv ` D ) = (Inv ` D )
17	10	ad2antrr 762	. . . . 5 |- ( ( ( ph /\ A e. ( F I G ) ) /\ x e. B ) -> D e. Cat )
18		fuciso.b	. . . . . . . 8 |- B = ( Base ` C )
19		relfunc 16522	. . . . . . . . 9 |- Rel ( C Func D )
20		1st2ndbr 7217	. . . . . . . . 9 |- ( ( Rel ( C Func D ) /\ F e. ( C Func D ) ) -> ( 1st ` F ) ( C Func D ) ( 2nd ` F ) )
21	19, 6, 20	sylancr 695	. . . . . . . 8 |- ( ph -> ( 1st ` F ) ( C Func D ) ( 2nd ` F ) )
22	18, 15, 21	funcf1 16526	. . . . . . 7 |- ( ph -> ( 1st ` F ) : B --> ( Base ` D ) )
23	22	adantr 481	. . . . . 6 |- ( ( ph /\ A e. ( F I G ) ) -> ( 1st ` F ) : B --> ( Base ` D ) )
24	23	ffvelrnda 6359	. . . . 5 |- ( ( ( ph /\ A e. ( F I G ) ) /\ x e. B ) -> ( ( 1st ` F ) ` x ) e. ( Base ` D ) )
25		1st2ndbr 7217	. . . . . . . . 9 |- ( ( Rel ( C Func D ) /\ G e. ( C Func D ) ) -> ( 1st ` G ) ( C Func D ) ( 2nd ` G ) )
26	19, 12, 25	sylancr 695	. . . . . . . 8 |- ( ph -> ( 1st ` G ) ( C Func D ) ( 2nd ` G ) )
27	18, 15, 26	funcf1 16526	. . . . . . 7 |- ( ph -> ( 1st ` G ) : B --> ( Base ` D ) )
28	27	adantr 481	. . . . . 6 |- ( ( ph /\ A e. ( F I G ) ) -> ( 1st ` G ) : B --> ( Base ` D ) )
29	28	ffvelrnda 6359	. . . . 5 |- ( ( ( ph /\ A e. ( F I G ) ) /\ x e. B ) -> ( ( 1st ` G ) ` x ) e. ( Base ` D ) )
30		fuciso.j	. . . . 5 |- J = ( Iso ` D )
31		eqid 2622	. . . . . . . . . . . 12 |- (Inv ` Q ) = (Inv ` Q )
32	2, 31, 11, 6, 12, 5	isoval 16425	. . . . . . . . . . 11 |- ( ph -> ( F I G ) = dom ( F (Inv ` Q ) G ) )
33	32	eleq2d 2687	. . . . . . . . . 10 |- ( ph -> ( A e. ( F I G ) <-> A e. dom ( F (Inv ` Q ) G ) ) )
34	2, 31, 11, 6, 12	invfun 16424	. . . . . . . . . . 11 |- ( ph -> Fun ( F (Inv ` Q ) G ) )
35		funfvbrb 6330	. . . . . . . . . . 11 |- ( Fun ( F (Inv ` Q ) G ) -> ( A e. dom ( F (Inv ` Q ) G ) <-> A ( F (Inv ` Q ) G ) ( ( F (Inv ` Q ) G ) ` A ) ) )
36	34, 35	syl 17	. . . . . . . . . 10 |- ( ph -> ( A e. dom ( F (Inv ` Q ) G ) <-> A ( F (Inv ` Q ) G ) ( ( F (Inv ` Q ) G ) ` A ) ) )
37	33, 36	bitrd 268	. . . . . . . . 9 |- ( ph -> ( A e. ( F I G ) <-> A ( F (Inv ` Q ) G ) ( ( F (Inv ` Q ) G ) ` A ) ) )
38	37	biimpa 501	. . . . . . . 8 |- ( ( ph /\ A e. ( F I G ) ) -> A ( F (Inv ` Q ) G ) ( ( F (Inv ` Q ) G ) ` A ) )
39	1, 18, 3, 6, 12, 31, 16	fucinv 16633	. . . . . . . . 9 |- ( ph -> ( A ( F (Inv ` Q ) G ) ( ( F (Inv ` Q ) G ) ` A ) <-> ( A e. ( F N G ) /\ ( ( F (Inv ` Q ) G ) ` A ) e. ( G N F ) /\ A. x e. B ( A ` x ) ( ( ( 1st ` F ) ` x ) (Inv ` D ) ( ( 1st ` G ) ` x ) ) ( ( ( F (Inv ` Q ) G ) ` A ) ` x ) ) ) )
40	39	adantr 481	. . . . . . . 8 |- ( ( ph /\ A e. ( F I G ) ) -> ( A ( F (Inv ` Q ) G ) ( ( F (Inv ` Q ) G ) ` A ) <-> ( A e. ( F N G ) /\ ( ( F (Inv ` Q ) G ) ` A ) e. ( G N F ) /\ A. x e. B ( A ` x ) ( ( ( 1st ` F ) ` x ) (Inv ` D ) ( ( 1st ` G ) ` x ) ) ( ( ( F (Inv ` Q ) G ) ` A ) ` x ) ) ) )
41	38, 40	mpbid 222	. . . . . . 7 |- ( ( ph /\ A e. ( F I G ) ) -> ( A e. ( F N G ) /\ ( ( F (Inv ` Q ) G ) ` A ) e. ( G N F ) /\ A. x e. B ( A ` x ) ( ( ( 1st ` F ) ` x ) (Inv ` D ) ( ( 1st ` G ) ` x ) ) ( ( ( F (Inv ` Q ) G ) ` A ) ` x ) ) )
42	41	simp3d 1075	. . . . . 6 |- ( ( ph /\ A e. ( F I G ) ) -> A. x e. B ( A ` x ) ( ( ( 1st ` F ) ` x ) (Inv ` D ) ( ( 1st ` G ) ` x ) ) ( ( ( F (Inv ` Q ) G ) ` A ) ` x ) )
43	42	r19.21bi 2932	. . . . 5 |- ( ( ( ph /\ A e. ( F I G ) ) /\ x e. B ) -> ( A ` x ) ( ( ( 1st ` F ) ` x ) (Inv ` D ) ( ( 1st ` G ) ` x ) ) ( ( ( F (Inv ` Q ) G ) ` A ) ` x ) )
44	15, 16, 17, 24, 29, 30, 43	inviso1 16426	. . . 4 |- ( ( ( ph /\ A e. ( F I G ) ) /\ x e. B ) -> ( A ` x ) e. ( ( ( 1st ` F ) ` x ) J ( ( 1st ` G ) ` x ) ) )
45	44	ralrimiva 2966	. . 3 |- ( ( ph /\ A e. ( F I G ) ) -> A. x e. B ( A ` x ) e. ( ( ( 1st ` F ) ` x ) J ( ( 1st ` G ) ` x ) ) )
46	14, 45	jca 554	. 2 |- ( ( ph /\ A e. ( F I G ) ) -> ( A e. ( F N G ) /\ A. x e. B ( A ` x ) e. ( ( ( 1st ` F ) ` x ) J ( ( 1st ` G ) ` x ) ) ) )
47	11	adantr 481	. . 3 |- ( ( ph /\ ( A e. ( F N G ) /\ A. x e. B ( A ` x ) e. ( ( ( 1st ` F ) ` x ) J ( ( 1st ` G ) ` x ) ) ) ) -> Q e. Cat )
48	6	adantr 481	. . 3 |- ( ( ph /\ ( A e. ( F N G ) /\ A. x e. B ( A ` x ) e. ( ( ( 1st ` F ) ` x ) J ( ( 1st ` G ) ` x ) ) ) ) -> F e. ( C Func D ) )
49	12	adantr 481	. . 3 |- ( ( ph /\ ( A e. ( F N G ) /\ A. x e. B ( A ` x ) e. ( ( ( 1st ` F ) ` x ) J ( ( 1st ` G ) ` x ) ) ) ) -> G e. ( C Func D ) )
50		simprl 794	. . . 4 |- ( ( ph /\ ( A e. ( F N G ) /\ A. x e. B ( A ` x ) e. ( ( ( 1st ` F ) ` x ) J ( ( 1st ` G ) ` x ) ) ) ) -> A e. ( F N G ) )
51		simprr 796	. . . . . . 7 |- ( ( ph /\ ( A e. ( F N G ) /\ A. x e. B ( A ` x ) e. ( ( ( 1st ` F ) ` x ) J ( ( 1st ` G ) ` x ) ) ) ) -> A. x e. B ( A ` x ) e. ( ( ( 1st ` F ) ` x ) J ( ( 1st ` G ) ` x ) ) )
52		fveq2 6191	. . . . . . . . 9 |- ( x = y -> ( A ` x ) = ( A ` y ) )
53		fveq2 6191	. . . . . . . . . 10 |- ( x = y -> ( ( 1st ` F ) ` x ) = ( ( 1st ` F ) ` y ) )
54		fveq2 6191	. . . . . . . . . 10 |- ( x = y -> ( ( 1st ` G ) ` x ) = ( ( 1st ` G ) ` y ) )
55	53, 54	oveq12d 6668	. . . . . . . . 9 |- ( x = y -> ( ( ( 1st ` F ) ` x ) J ( ( 1st ` G ) ` x ) ) = ( ( ( 1st ` F ) ` y ) J ( ( 1st ` G ) ` y ) ) )
56	52, 55	eleq12d 2695	. . . . . . . 8 |- ( x = y -> ( ( A ` x ) e. ( ( ( 1st ` F ) ` x ) J ( ( 1st ` G ) ` x ) ) <-> ( A ` y ) e. ( ( ( 1st ` F ) ` y ) J ( ( 1st ` G ) ` y ) ) ) )
57	56	rspccva 3308	. . . . . . 7 |- ( ( A. x e. B ( A ` x ) e. ( ( ( 1st ` F ) ` x ) J ( ( 1st ` G ) ` x ) ) /\ y e. B ) -> ( A ` y ) e. ( ( ( 1st ` F ) ` y ) J ( ( 1st ` G ) ` y ) ) )
58	51, 57	sylan 488	. . . . . 6 |- ( ( ( ph /\ ( A e. ( F N G ) /\ A. x e. B ( A ` x ) e. ( ( ( 1st ` F ) ` x ) J ( ( 1st ` G ) ` x ) ) ) ) /\ y e. B ) -> ( A ` y ) e. ( ( ( 1st ` F ) ` y ) J ( ( 1st ` G ) ` y ) ) )
59	10	ad2antrr 762	. . . . . . 7 |- ( ( ( ph /\ ( A e. ( F N G ) /\ A. x e. B ( A ` x ) e. ( ( ( 1st ` F ) ` x ) J ( ( 1st ` G ) ` x ) ) ) ) /\ y e. B ) -> D e. Cat )
60	22	adantr 481	. . . . . . . 8 |- ( ( ph /\ ( A e. ( F N G ) /\ A. x e. B ( A ` x ) e. ( ( ( 1st ` F ) ` x ) J ( ( 1st ` G ) ` x ) ) ) ) -> ( 1st ` F ) : B --> ( Base ` D ) )
61	60	ffvelrnda 6359	. . . . . . 7 |- ( ( ( ph /\ ( A e. ( F N G ) /\ A. x e. B ( A ` x ) e. ( ( ( 1st ` F ) ` x ) J ( ( 1st ` G ) ` x ) ) ) ) /\ y e. B ) -> ( ( 1st ` F ) ` y ) e. ( Base ` D ) )
62	27	adantr 481	. . . . . . . 8 |- ( ( ph /\ ( A e. ( F N G ) /\ A. x e. B ( A ` x ) e. ( ( ( 1st ` F ) ` x ) J ( ( 1st ` G ) ` x ) ) ) ) -> ( 1st ` G ) : B --> ( Base ` D ) )
63	62	ffvelrnda 6359	. . . . . . 7 |- ( ( ( ph /\ ( A e. ( F N G ) /\ A. x e. B ( A ` x ) e. ( ( ( 1st ` F ) ` x ) J ( ( 1st ` G ) ` x ) ) ) ) /\ y e. B ) -> ( ( 1st ` G ) ` y ) e. ( Base ` D ) )
64	15, 16, 59, 61, 63, 30	isoval 16425	. . . . . 6 |- ( ( ( ph /\ ( A e. ( F N G ) /\ A. x e. B ( A ` x ) e. ( ( ( 1st ` F ) ` x ) J ( ( 1st ` G ) ` x ) ) ) ) /\ y e. B ) -> ( ( ( 1st ` F ) ` y ) J ( ( 1st ` G ) ` y ) ) = dom ( ( ( 1st ` F ) ` y ) (Inv ` D ) ( ( 1st ` G ) ` y ) ) )
65	58, 64	eleqtrd 2703	. . . . 5 |- ( ( ( ph /\ ( A e. ( F N G ) /\ A. x e. B ( A ` x ) e. ( ( ( 1st ` F ) ` x ) J ( ( 1st ` G ) ` x ) ) ) ) /\ y e. B ) -> ( A ` y ) e. dom ( ( ( 1st ` F ) ` y ) (Inv ` D ) ( ( 1st ` G ) ` y ) ) )
66	15, 16, 59, 61, 63	invfun 16424	. . . . . 6 |- ( ( ( ph /\ ( A e. ( F N G ) /\ A. x e. B ( A ` x ) e. ( ( ( 1st ` F ) ` x ) J ( ( 1st ` G ) ` x ) ) ) ) /\ y e. B ) -> Fun ( ( ( 1st ` F ) ` y ) (Inv ` D ) ( ( 1st ` G ) ` y ) ) )
67		funfvbrb 6330	. . . . . 6 |- ( Fun ( ( ( 1st ` F ) ` y ) (Inv ` D ) ( ( 1st ` G ) ` y ) ) -> ( ( A ` y ) e. dom ( ( ( 1st ` F ) ` y ) (Inv ` D ) ( ( 1st ` G ) ` y ) ) <-> ( A ` y ) ( ( ( 1st ` F ) ` y ) (Inv ` D ) ( ( 1st ` G ) ` y ) ) ( ( ( ( 1st ` F ) ` y ) (Inv ` D ) ( ( 1st ` G ) ` y ) ) ` ( A ` y ) ) ) )
68	66, 67	syl 17	. . . . 5 |- ( ( ( ph /\ ( A e. ( F N G ) /\ A. x e. B ( A ` x ) e. ( ( ( 1st ` F ) ` x ) J ( ( 1st ` G ) ` x ) ) ) ) /\ y e. B ) -> ( ( A ` y ) e. dom ( ( ( 1st ` F ) ` y ) (Inv ` D ) ( ( 1st ` G ) ` y ) ) <-> ( A ` y ) ( ( ( 1st ` F ) ` y ) (Inv ` D ) ( ( 1st ` G ) ` y ) ) ( ( ( ( 1st ` F ) ` y ) (Inv ` D ) ( ( 1st ` G ) ` y ) ) ` ( A ` y ) ) ) )
69	65, 68	mpbid 222	. . . 4 |- ( ( ( ph /\ ( A e. ( F N G ) /\ A. x e. B ( A ` x ) e. ( ( ( 1st ` F ) ` x ) J ( ( 1st ` G ) ` x ) ) ) ) /\ y e. B ) -> ( A ` y ) ( ( ( 1st ` F ) ` y ) (Inv ` D ) ( ( 1st ` G ) ` y ) ) ( ( ( ( 1st ` F ) ` y ) (Inv ` D ) ( ( 1st ` G ) ` y ) ) ` ( A ` y ) ) )
70	1, 18, 3, 48, 49, 31, 16, 50, 69	invfuc 16634	. . 3 |- ( ( ph /\ ( A e. ( F N G ) /\ A. x e. B ( A ` x ) e. ( ( ( 1st ` F ) ` x ) J ( ( 1st ` G ) ` x ) ) ) ) -> A ( F (Inv ` Q ) G ) ( y e. B |-> ( ( ( ( 1st ` F ) ` y ) (Inv ` D ) ( ( 1st ` G ) ` y ) ) ` ( A ` y ) ) ) )
71	2, 31, 47, 48, 49, 5, 70	inviso1 16426	. 2 |- ( ( ph /\ ( A e. ( F N G ) /\ A. x e. B ( A ` x ) e. ( ( ( 1st ` F ) ` x ) J ( ( 1st ` G ) ` x ) ) ) ) -> A e. ( F I G ) )
72	46, 71	impbida 877	1 |- ( ph -> ( A e. ( F I G ) <-> ( A e. ( F N G ) /\ A. x e. B ( A ` x ) e. ( ( ( 1st ` F ) ` x ) J ( ( 1st ` G ) ` x ) ) ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   class class class wbr 4653   |-> cmpt 4729   dom cdm 5114   Rel wrel 5119   Fun wfun 5882   -->wf 5884   ` cfv 5888  (class class class)co 6650   1stc1st 7166   2ndc2nd 7167   Basecbs 15857   Catccat 16325  Invcinv 16405   Iso ciso 16406   Func cfunc 16514   Nat cnat 16601   FuncCat cfuc 16602

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  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  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-nel 2898  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-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  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-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  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-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-map 7859  df-ixp 7909  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-dec 11494  df-uz 11688  df-fz 12327  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-hom 15966  df-cco 15967  df-cat 16329  df-cid 16330  df-sect 16407  df-inv 16408  df-iso 16409  df-func 16518  df-nat 16603  df-fuc 16604

This theorem is referenced by:  yonffthlem  16922