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Theorem fullfo 15814
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 15813 . . . 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 466 . . 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 17 . 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 467 . . . 4  |-  ( (
ph  /\  x  =  X )  ->  Y  e.  B )
11 simplr 761 . . . . . 6  |-  ( ( ( ph  /\  x  =  X )  /\  y  =  Y )  ->  x  =  X )
12 simpr 463 . . . . . 6  |-  ( ( ( ph  /\  x  =  X )  /\  y  =  Y )  ->  y  =  Y )
1311, 12oveq12d 6322 . . . . 5  |-  ( ( ( ph  /\  x  =  X )  /\  y  =  Y )  ->  (
x G y )  =  ( X G Y ) )
1411, 12oveq12d 6322 . . . . 5  |-  ( ( ( ph  /\  x  =  X )  /\  y  =  Y )  ->  (
x H y )  =  ( X H Y ) )
1511fveq2d 5884 . . . . . 6  |-  ( ( ( ph  /\  x  =  X )  /\  y  =  Y )  ->  ( F `  x )  =  ( F `  X ) )
1612fveq2d 5884 . . . . . 6  |-  ( ( ( ph  /\  x  =  X )  /\  y  =  Y )  ->  ( F `  y )  =  ( F `  Y ) )
1715, 16oveq12d 6322 . . . . 5  |-  ( ( ( ph  /\  x  =  X )  /\  y  =  Y )  ->  (
( F `  x
) J ( F `
 y ) )  =  ( ( F `
 X ) J ( F `  Y
) ) )
1813, 14, 17foeq123d 5826 . . . 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 3186 . . 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 3184 . 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 371    = wceq 1438    e. wcel 1869   A.wral 2776   class class class wbr 4422   -onto->wfo 5598   ` cfv 5600  (class class class)co 6304   Basecbs 15118   Hom chom 15198    Func cfunc 15756   Full cful 15804
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1666  ax-4 1679  ax-5 1749  ax-6 1795  ax-7 1840  ax-8 1871  ax-9 1873  ax-10 1888  ax-11 1893  ax-12 1906  ax-13 2054  ax-ext 2401  ax-rep 4535  ax-sep 4545  ax-nul 4554  ax-pow 4601  ax-pr 4659  ax-un 6596
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 985  df-tru 1441  df-ex 1661  df-nf 1665  df-sb 1788  df-eu 2270  df-mo 2271  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2573  df-ne 2621  df-ral 2781  df-rex 2782  df-reu 2783  df-rab 2785  df-v 3084  df-sbc 3302  df-csb 3398  df-dif 3441  df-un 3443  df-in 3445  df-ss 3452  df-nul 3764  df-if 3912  df-pw 3983  df-sn 3999  df-pr 4001  df-op 4005  df-uni 4219  df-iun 4300  df-br 4423  df-opab 4482  df-mpt 4483  df-id 4767  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-iota 5564  df-fun 5602  df-fn 5603  df-f 5604  df-f1 5605  df-fo 5606  df-f1o 5607  df-fv 5608  df-ov 6307  df-oprab 6308  df-mpt2 6309  df-1st 6806  df-2nd 6807  df-map 7484  df-ixp 7533  df-func 15760  df-full 15806
This theorem is referenced by:  fulli  15815  ffthf1o  15821  fulloppc  15824  cofull  15836
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