Theorem isfull2 16571

Description: Equivalent condition for a full functor. (Contributed by Mario Carneiro, 27-Jan-2017.)

Hypotheses

Ref	Expression
isfull.b	|- B = ( Base ` C )
isfull.j	|- J = ( Hom ` D )
isfull.h	|- H = ( Hom ` C )

Assertion

Ref	Expression
isfull2	|- ( F ( C Full D ) G <-> ( F ( C Func D ) G /\ A. x e. B A. y e. B ( x G y ) : ( x H y ) -onto-> ( ( F ` x ) J ( F ` y ) ) ) )

Distinct variable groups:   x, y, B   x, C, y   x, D, y   x, H, y   x, J, y   x, F, y   x, G, y

Proof of Theorem isfull2

Step	Hyp	Ref	Expression
1		isfull.b	. . 3 |- B = ( Base ` C )
2		isfull.j	. . 3 |- J = ( Hom ` D )
3	1, 2	isfull 16570	. 2 |- ( F ( C Full D ) G <-> ( F ( C Func D ) G /\ A. x e. B A. y e. B ran ( x G y ) = ( ( F ` x ) J ( F ` y ) ) ) )
4		isfull.h	. . . . . . 7 |- H = ( Hom ` C )
5		simpll 790	. . . . . . 7 |- ( ( ( F ( C Func D ) G /\ x e. B ) /\ y e. B ) -> F ( C Func D ) G )
6		simplr 792	. . . . . . 7 |- ( ( ( F ( C Func D ) G /\ x e. B ) /\ y e. B ) -> x e. B )
7		simpr 477	. . . . . . 7 |- ( ( ( F ( C Func D ) G /\ x e. B ) /\ y e. B ) -> y e. B )
8	1, 4, 2, 5, 6, 7	funcf2 16528	. . . . . 6 |- ( ( ( F ( C Func D ) G /\ x e. B ) /\ y e. B ) -> ( x G y ) : ( x H y ) --> ( ( F ` x ) J ( F ` y ) ) )
9		ffn 6045	. . . . . 6 |- ( ( x G y ) : ( x H y ) --> ( ( F ` x ) J ( F ` y ) ) -> ( x G y ) Fn ( x H y ) )
10		df-fo 5894	. . . . . . 7 |- ( ( x G y ) : ( x H y ) -onto-> ( ( F ` x ) J ( F ` y ) ) <-> ( ( x G y ) Fn ( x H y ) /\ ran ( x G y ) = ( ( F ` x ) J ( F ` y ) ) ) )
11	10	baib 944	. . . . . 6 |- ( ( x G y ) Fn ( x H y ) -> ( ( x G y ) : ( x H y ) -onto-> ( ( F ` x ) J ( F ` y ) ) <-> ran ( x G y ) = ( ( F ` x ) J ( F ` y ) ) ) )
12	8, 9, 11	3syl 18	. . . . 5 |- ( ( ( F ( C Func D ) G /\ x e. B ) /\ y e. B ) -> ( ( x G y ) : ( x H y ) -onto-> ( ( F ` x ) J ( F ` y ) ) <-> ran ( x G y ) = ( ( F ` x ) J ( F ` y ) ) ) )
13	12	ralbidva 2985	. . . 4 |- ( ( F ( C Func D ) G /\ x e. B ) -> ( A. y e. B ( x G y ) : ( x H y ) -onto-> ( ( F ` x ) J ( F ` y ) ) <-> A. y e. B ran ( x G y ) = ( ( F ` x ) J ( F ` y ) ) ) )
14	13	ralbidva 2985	. . 3 |- ( F ( C Func D ) G -> ( A. x e. B A. y e. B ( x G y ) : ( x H y ) -onto-> ( ( F ` x ) J ( F ` y ) ) <-> A. x e. B A. y e. B ran ( x G y ) = ( ( F ` x ) J ( F ` y ) ) ) )
15	14	pm5.32i 669	. 2 |- ( ( F ( C Func D ) G /\ A. x e. B A. y e. B ( x G y ) : ( x H y ) -onto-> ( ( F ` x ) J ( F ` y ) ) ) <-> ( F ( C Func D ) G /\ A. x e. B A. y e. B ran ( x G y ) = ( ( F ` x ) J ( F ` y ) ) ) )
16	3, 15	bitr4i 267	1 |- ( F ( C Full D ) G <-> ( F ( C Func D ) G /\ A. x e. B A. y e. B ( x G y ) : ( x H y ) -onto-> ( ( F ` x ) J ( F ` y ) ) ) )

Syntax hints:   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   class class class wbr 4653   ran crn 5115   Fn wfn 5883   -->wf 5884   -onto->wfo 5886   ` cfv 5888  (class class class)co 6650   Basecbs 15857   Hom chom 15952   Func cfunc 16514   Full cful 16562

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-map 7859  df-ixp 7909  df-func 16518  df-full 16564

This theorem is referenced by:  fullfo  16572  isffth2  16576  cofull  16594  fullestrcsetc  16791  fullsetcestrc  16806