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Theorem fullfo 14036
Description: The morphism map of a full functor is a surjection. (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
)
fullfo.f  |-  ( ph  ->  F ( C Full  D
) G )
fullfo.x  |-  ( ph  ->  X  e.  B )
fullfo.y  |-  ( ph  ->  Y  e.  B )
Assertion
Ref Expression
fullfo  |-  ( ph  ->  ( X G Y ) : ( X H Y ) -onto-> ( ( F `  X
) J ( F `
 Y ) ) )

Proof of Theorem fullfo
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fullfo.f . . 3  |-  ( ph  ->  F ( C Full  D
) G )
2 isfull.b . . . . 5  |-  B  =  ( Base `  C
)
3 isfull.j . . . . 5  |-  J  =  (  Hom  `  D
)
4 isfull.h . . . . 5  |-  H  =  (  Hom  `  C
)
52, 3, 4isfull2 14035 . . . 4  |-  ( 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 ) ) ) )
65simprbi 451 . . 3  |-  ( F ( C Full  D ) G  ->  A. x  e.  B  A. y  e.  B  ( x G y ) : ( x H y ) -onto-> ( ( F `
 x ) J ( F `  y
) ) )
71, 6syl 16 . 2  |-  ( ph  ->  A. x  e.  B  A. y  e.  B  ( x G y ) : ( x H y ) -onto-> ( ( F `  x
) J ( F `
 y ) ) )
8 fullfo.x . . 3  |-  ( ph  ->  X  e.  B )
9 fullfo.y . . . . 5  |-  ( ph  ->  Y  e.  B )
109adantr 452 . . . 4  |-  ( (
ph  /\  x  =  X )  ->  Y  e.  B )
11 simplr 732 . . . . . 6  |-  ( ( ( ph  /\  x  =  X )  /\  y  =  Y )  ->  x  =  X )
12 simpr 448 . . . . . 6  |-  ( ( ( ph  /\  x  =  X )  /\  y  =  Y )  ->  y  =  Y )
1311, 12oveq12d 6038 . . . . 5  |-  ( ( ( ph  /\  x  =  X )  /\  y  =  Y )  ->  (
x G y )  =  ( X G Y ) )
1411, 12oveq12d 6038 . . . . 5  |-  ( ( ( ph  /\  x  =  X )  /\  y  =  Y )  ->  (
x H y )  =  ( X H Y ) )
1511fveq2d 5672 . . . . . 6  |-  ( ( ( ph  /\  x  =  X )  /\  y  =  Y )  ->  ( F `  x )  =  ( F `  X ) )
1612fveq2d 5672 . . . . . 6  |-  ( ( ( ph  /\  x  =  X )  /\  y  =  Y )  ->  ( F `  y )  =  ( F `  Y ) )
1715, 16oveq12d 6038 . . . . 5  |-  ( ( ( ph  /\  x  =  X )  /\  y  =  Y )  ->  (
( F `  x
) J ( F `
 y ) )  =  ( ( F `
 X ) J ( F `  Y
) ) )
1813, 14, 17foeq123d 5610 . . . 4  |-  ( ( ( ph  /\  x  =  X )  /\  y  =  Y )  ->  (
( x G y ) : ( x H y ) -onto-> ( ( F `  x
) J ( F `
 y ) )  <-> 
( X G Y ) : ( X H Y ) -onto-> ( ( F `  X
) J ( F `
 Y ) ) ) )
1910, 18rspcdv 2998 . . 3  |-  ( (
ph  /\  x  =  X )  ->  ( A. y  e.  B  ( x G y ) : ( x H y ) -onto-> ( ( F `  x
) J ( F `
 y ) )  ->  ( X G Y ) : ( X H Y )
-onto-> ( ( F `  X ) J ( F `  Y ) ) ) )
208, 19rspcimdv 2996 . 2  |-  ( ph  ->  ( A. x  e.  B  A. y  e.  B  ( x G y ) : ( x H y )
-onto-> ( ( F `  x ) J ( F `  y ) )  ->  ( X G Y ) : ( X H Y )
-onto-> ( ( F `  X ) J ( F `  Y ) ) ) )
217, 20mpd 15 1  |-  ( ph  ->  ( X G Y ) : ( X H Y ) -onto-> ( ( F `  X
) J ( F `
 Y ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1717   A.wral 2649   class class class wbr 4153   -onto->wfo 5392   ` cfv 5394  (class class class)co 6020   Basecbs 13396    Hom chom 13467    Func cfunc 13978   Full cful 14026
This theorem is referenced by:  fulli  14037  ffthf1o  14043  fulloppc  14046  cofull  14058
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-rep 4261  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-reu 2656  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-iun 4037  df-br 4154  df-opab 4208  df-mpt 4209  df-id 4439  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-1st 6288  df-2nd 6289  df-map 6956  df-ixp 7000  df-func 13982  df-full 14028
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