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Theorem ffthf1o 13793
Description: The morphism map of a fully faithful functor is a bijection. (Contributed by Mario Carneiro, 29-Jan-2017.)
Hypotheses
Ref Expression
isfth.b  |-  B  =  ( Base `  C
)
isfth.h  |-  H  =  (  Hom  `  C
)
isfth.j  |-  J  =  (  Hom  `  D
)
ffthf1o.f  |-  ( ph  ->  F ( ( C Full 
D )  i^i  ( C Faith  D ) ) G )
ffthf1o.x  |-  ( ph  ->  X  e.  B )
ffthf1o.y  |-  ( ph  ->  Y  e.  B )
Assertion
Ref Expression
ffthf1o  |-  ( ph  ->  ( X G Y ) : ( X H Y ) -1-1-onto-> ( ( F `  X ) J ( F `  Y ) ) )

Proof of Theorem ffthf1o
StepHypRef Expression
1 isfth.b . . 3  |-  B  =  ( Base `  C
)
2 isfth.h . . 3  |-  H  =  (  Hom  `  C
)
3 isfth.j . . 3  |-  J  =  (  Hom  `  D
)
4 ffthf1o.f . . . . 5  |-  ( ph  ->  F ( ( C Full 
D )  i^i  ( C Faith  D ) ) G )
5 brin 4070 . . . . 5  |-  ( F ( ( C Full  D
)  i^i  ( C Faith  D ) ) G  <->  ( F
( C Full  D ) G  /\  F ( C Faith 
D ) G ) )
64, 5sylib 188 . . . 4  |-  ( ph  ->  ( F ( C Full 
D ) G  /\  F ( C Faith  D
) G ) )
76simprd 449 . . 3  |-  ( ph  ->  F ( C Faith  D
) G )
8 ffthf1o.x . . 3  |-  ( ph  ->  X  e.  B )
9 ffthf1o.y . . 3  |-  ( ph  ->  Y  e.  B )
101, 2, 3, 7, 8, 9fthf1 13791 . 2  |-  ( ph  ->  ( X G Y ) : ( X H Y ) -1-1-> ( ( F `  X
) J ( F `
 Y ) ) )
116simpld 445 . . 3  |-  ( ph  ->  F ( C Full  D
) G )
121, 3, 2, 11, 8, 9fullfo 13786 . 2  |-  ( ph  ->  ( X G Y ) : ( X H Y ) -onto-> ( ( F `  X
) J ( F `
 Y ) ) )
13 df-f1o 5262 . 2  |-  ( ( X G Y ) : ( X H Y ) -1-1-onto-> ( ( F `  X ) J ( F `  Y ) )  <->  ( ( X G Y ) : ( X H Y ) -1-1-> ( ( F `
 X ) J ( F `  Y
) )  /\  ( X G Y ) : ( X H Y ) -onto-> ( ( F `
 X ) J ( F `  Y
) ) ) )
1410, 12, 13sylanbrc 645 1  |-  ( ph  ->  ( X G Y ) : ( X H Y ) -1-1-onto-> ( ( F `  X ) J ( F `  Y ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684    i^i cin 3151   class class class wbr 4023   -1-1->wf1 5252   -onto->wfo 5253   -1-1-onto->wf1o 5254   ` cfv 5255  (class class class)co 5858   Basecbs 13148    Hom chom 13219   Full cful 13776   Faith cfth 13777
This theorem is referenced by:  catcisolem  13938
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-map 6774  df-ixp 6818  df-func 13732  df-full 13778  df-fth 13779
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