MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  fullfo Structured version   Unicode version

Theorem fullfo 14926
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 14925 . . . 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 464 . . 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 465 . . . 4  |-  ( (
ph  /\  x  =  X )  ->  Y  e.  B )
11 simplr 754 . . . . . 6  |-  ( ( ( ph  /\  x  =  X )  /\  y  =  Y )  ->  x  =  X )
12 simpr 461 . . . . . 6  |-  ( ( ( ph  /\  x  =  X )  /\  y  =  Y )  ->  y  =  Y )
1311, 12oveq12d 6210 . . . . 5  |-  ( ( ( ph  /\  x  =  X )  /\  y  =  Y )  ->  (
x G y )  =  ( X G Y ) )
1411, 12oveq12d 6210 . . . . 5  |-  ( ( ( ph  /\  x  =  X )  /\  y  =  Y )  ->  (
x H y )  =  ( X H Y ) )
1511fveq2d 5795 . . . . . 6  |-  ( ( ( ph  /\  x  =  X )  /\  y  =  Y )  ->  ( F `  x )  =  ( F `  X ) )
1612fveq2d 5795 . . . . . 6  |-  ( ( ( ph  /\  x  =  X )  /\  y  =  Y )  ->  ( F `  y )  =  ( F `  Y ) )
1715, 16oveq12d 6210 . . . . 5  |-  ( ( ( ph  /\  x  =  X )  /\  y  =  Y )  ->  (
( F `  x
) J ( F `
 y ) )  =  ( ( F `
 X ) J ( F `  Y
) ) )
1813, 14, 17foeq123d 5737 . . . 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 3174 . . 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 3172 . 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 setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1758   A.wral 2795   class class class wbr 4392   -onto->wfo 5516   ` cfv 5518  (class class class)co 6192   Basecbs 14278   Hom chom 14353    Func cfunc 14868   Full cful 14916
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4503  ax-sep 4513  ax-nul 4521  ax-pow 4570  ax-pr 4631  ax-un 6474
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-reu 2802  df-rab 2804  df-v 3072  df-sbc 3287  df-csb 3389  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-nul 3738  df-if 3892  df-pw 3962  df-sn 3978  df-pr 3980  df-op 3984  df-uni 4192  df-iun 4273  df-br 4393  df-opab 4451  df-mpt 4452  df-id 4736  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5481  df-fun 5520  df-fn 5521  df-f 5522  df-f1 5523  df-fo 5524  df-f1o 5525  df-fv 5526  df-ov 6195  df-oprab 6196  df-mpt2 6197  df-1st 6679  df-2nd 6680  df-map 7318  df-ixp 7366  df-func 14872  df-full 14918
This theorem is referenced by:  fulli  14927  ffthf1o  14933  fulloppc  14936  cofull  14948
  Copyright terms: Public domain W3C validator