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Theorem ffthf1o 14036
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 4193 . . . . 5  |-  ( F ( ( C Full  D
)  i^i  ( C Faith  D ) ) G  <->  ( F
( C Full  D ) G  /\  F ( C Faith 
D ) G ) )
64, 5sylib 189 . . . 4  |-  ( ph  ->  ( F ( C Full 
D ) G  /\  F ( C Faith  D
) G ) )
76simprd 450 . . 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 14034 . 2  |-  ( ph  ->  ( X G Y ) : ( X H Y ) -1-1-> ( ( F `  X
) J ( F `
 Y ) ) )
116simpld 446 . . 3  |-  ( ph  ->  F ( C Full  D
) G )
121, 3, 2, 11, 8, 9fullfo 14029 . 2  |-  ( ph  ->  ( X G Y ) : ( X H Y ) -onto-> ( ( F `  X
) J ( F `
 Y ) ) )
13 df-f1o 5394 . 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 646 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 359    = wceq 1649    e. wcel 1717    i^i cin 3255   class class class wbr 4146   -1-1->wf1 5384   -onto->wfo 5385   -1-1-onto->wf1o 5386   ` cfv 5387  (class class class)co 6013   Basecbs 13389    Hom chom 13460   Full cful 14019   Faith cfth 14020
This theorem is referenced by:  catcisolem  14181
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 2361  ax-rep 4254  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337  ax-un 4634
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 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-ral 2647  df-rex 2648  df-reu 2649  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-op 3759  df-uni 3951  df-iun 4030  df-br 4147  df-opab 4201  df-mpt 4202  df-id 4432  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-f1 5392  df-fo 5393  df-f1o 5394  df-fv 5395  df-ov 6016  df-oprab 6017  df-mpt2 6018  df-1st 6281  df-2nd 6282  df-map 6949  df-ixp 6993  df-func 13975  df-full 14021  df-fth 14022
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