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Theorem isfull2 13801
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 13800 . 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 . . . . . . . 8  |-  H  =  (  Hom  `  C
)
5 simpll 730 . . . . . . . 8  |-  ( ( ( F ( C 
Func  D ) G  /\  x  e.  B )  /\  y  e.  B
)  ->  F ( C  Func  D ) G )
6 simplr 731 . . . . . . . 8  |-  ( ( ( F ( C 
Func  D ) G  /\  x  e.  B )  /\  y  e.  B
)  ->  x  e.  B )
7 simpr 447 . . . . . . . 8  |-  ( ( ( F ( C 
Func  D ) G  /\  x  e.  B )  /\  y  e.  B
)  ->  y  e.  B )
81, 4, 2, 5, 6, 7funcf2 13758 . . . . . . 7  |-  ( ( ( F ( C 
Func  D ) G  /\  x  e.  B )  /\  y  e.  B
)  ->  ( x G y ) : ( x H y ) --> ( ( F `
 x ) J ( F `  y
) ) )
9 ffn 5405 . . . . . . 7  |-  ( ( x G y ) : ( x H y ) --> ( ( F `  x ) J ( F `  y ) )  -> 
( x G y )  Fn  ( x H y ) )
108, 9syl 15 . . . . . 6  |-  ( ( ( F ( C 
Func  D ) G  /\  x  e.  B )  /\  y  e.  B
)  ->  ( x G y )  Fn  ( x H y ) )
11 df-fo 5277 . . . . . . 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 ) ) ) )
1211baib 871 . . . . . 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 ) ) ) )
1310, 12syl 15 . . . . 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 ) ) ) )
1413ralbidva 2572 . . . 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 ) ) ) )
1514ralbidva 2572 . . 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 ) ) ) )
1615pm5.32i 618 . 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 ) ) ) )
173, 16bitr4i 243 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 176    /\ wa 358    = wceq 1632    e. wcel 1696   A.wral 2556   class class class wbr 4039   ran crn 4706    Fn wfn 5266   -->wf 5267   -onto->wfo 5269   ` cfv 5271  (class class class)co 5874   Basecbs 13164    Hom chom 13235    Func cfunc 13744   Full cful 13792
This theorem is referenced by:  fullfo  13802  isffth2  13806  cofull  13824
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-map 6790  df-ixp 6834  df-func 13748  df-full 13794
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