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Theorem fullfo 14101
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 14100 . . . 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 6091 . . . . 5  |-  ( ( ( ph  /\  x  =  X )  /\  y  =  Y )  ->  (
x G y )  =  ( X G Y ) )
1411, 12oveq12d 6091 . . . . 5  |-  ( ( ( ph  /\  x  =  X )  /\  y  =  Y )  ->  (
x H y )  =  ( X H Y ) )
1511fveq2d 5724 . . . . . 6  |-  ( ( ( ph  /\  x  =  X )  /\  y  =  Y )  ->  ( F `  x )  =  ( F `  X ) )
1612fveq2d 5724 . . . . . 6  |-  ( ( ( ph  /\  x  =  X )  /\  y  =  Y )  ->  ( F `  y )  =  ( F `  Y ) )
1715, 16oveq12d 6091 . . . . 5  |-  ( ( ( ph  /\  x  =  X )  /\  y  =  Y )  ->  (
( F `  x
) J ( F `
 y ) )  =  ( ( F `
 X ) J ( F `  Y
) ) )
1813, 14, 17foeq123d 5662 . . . 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 3047 . . 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 3045 . 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 1652    e. wcel 1725   A.wral 2697   class class class wbr 4204   -onto->wfo 5444   ` cfv 5446  (class class class)co 6073   Basecbs 13461    Hom chom 13532    Func cfunc 14043   Full cful 14091
This theorem is referenced by:  fulli  14102  ffthf1o  14108  fulloppc  14111  cofull  14123
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-map 7012  df-ixp 7056  df-func 14047  df-full 14093
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