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Theorem isfull2 14109
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
StepHypRef Expression
1 isfull.b . . 3  |-  B  =  ( Base `  C
)
2 isfull.j . . 3  |-  J  =  (  Hom  `  D
)
31, 2isfull 14108 . 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 732 . . . . . . 7  |-  ( ( ( F ( C 
Func  D ) G  /\  x  e.  B )  /\  y  e.  B
)  ->  F ( C  Func  D ) G )
6 simplr 733 . . . . . . 7  |-  ( ( ( F ( C 
Func  D ) G  /\  x  e.  B )  /\  y  e.  B
)  ->  x  e.  B )
7 simpr 449 . . . . . . 7  |-  ( ( ( F ( C 
Func  D ) G  /\  x  e.  B )  /\  y  e.  B
)  ->  y  e.  B )
81, 4, 2, 5, 6, 7funcf2 14066 . . . . . 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 5592 . . . . . 6  |-  ( ( x G y ) : ( x H y ) --> ( ( F `  x ) J ( F `  y ) )  -> 
( x G y )  Fn  ( x H y ) )
10 df-fo 5461 . . . . . . 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 ) ) ) )
1110baib 873 . . . . . 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 ) ) ) )
128, 9, 113syl 19 . . . . 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 ) ) ) )
1312ralbidva 2722 . . . 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 ) ) ) )
1413ralbidva 2722 . . 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 ) ) ) )
1514pm5.32i 620 . 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 ) ) ) )
163, 15bitr4i 245 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 ) ) ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 178    /\ wa 360    = wceq 1653    e. wcel 1726   A.wral 2706   class class class wbr 4213   ran crn 4880    Fn wfn 5450   -->wf 5451   -onto->wfo 5453   ` cfv 5455  (class class class)co 6082   Basecbs 13470    Hom chom 13541    Func cfunc 14052   Full cful 14100
This theorem is referenced by:  fullfo  14110  isffth2  14114  cofull  14132
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-rep 4321  ax-sep 4331  ax-nul 4339  ax-pow 4378  ax-pr 4404  ax-un 4702
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-ral 2711  df-rex 2712  df-reu 2713  df-rab 2715  df-v 2959  df-sbc 3163  df-csb 3253  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3630  df-if 3741  df-pw 3802  df-sn 3821  df-pr 3822  df-op 3824  df-uni 4017  df-iun 4096  df-br 4214  df-opab 4268  df-mpt 4269  df-id 4499  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-rn 4890  df-res 4891  df-ima 4892  df-iota 5419  df-fun 5457  df-fn 5458  df-f 5459  df-f1 5460  df-fo 5461  df-f1o 5462  df-fv 5463  df-ov 6085  df-oprab 6086  df-mpt2 6087  df-1st 6350  df-2nd 6351  df-map 7021  df-ixp 7065  df-func 14056  df-full 14102
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