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Theorem ffthf1o 14832
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 4344 . . . . 5  |-  ( F ( ( C Full  D
)  i^i  ( C Faith  D ) ) G  <->  ( F
( C Full  D ) G  /\  F ( C Faith 
D ) G ) )
64, 5sylib 196 . . . 4  |-  ( ph  ->  ( F ( C Full 
D ) G  /\  F ( C Faith  D
) G ) )
76simprd 463 . . 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 14830 . 2  |-  ( ph  ->  ( X G Y ) : ( X H Y ) -1-1-> ( ( F `  X
) J ( F `
 Y ) ) )
116simpld 459 . . 3  |-  ( ph  ->  F ( C Full  D
) G )
121, 3, 2, 11, 8, 9fullfo 14825 . 2  |-  ( ph  ->  ( X G Y ) : ( X H Y ) -onto-> ( ( F `  X
) J ( F `
 Y ) ) )
13 df-f1o 5428 . 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 664 1  |-  ( ph  ->  ( X G Y ) : ( X H Y ) -1-1-onto-> ( ( F `  X ) J ( F `  Y ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756    i^i cin 3330   class class class wbr 4295   -1-1->wf1 5418   -onto->wfo 5419   -1-1-onto->wf1o 5420   ` cfv 5421  (class class class)co 6094   Basecbs 14177   Hom chom 14252   Full cful 14815   Faith cfth 14816
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4406  ax-sep 4416  ax-nul 4424  ax-pow 4473  ax-pr 4534  ax-un 6375
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2571  df-ne 2611  df-ral 2723  df-rex 2724  df-reu 2725  df-rab 2727  df-v 2977  df-sbc 3190  df-csb 3292  df-dif 3334  df-un 3336  df-in 3338  df-ss 3345  df-nul 3641  df-if 3795  df-pw 3865  df-sn 3881  df-pr 3883  df-op 3887  df-uni 4095  df-iun 4176  df-br 4296  df-opab 4354  df-mpt 4355  df-id 4639  df-xp 4849  df-rel 4850  df-cnv 4851  df-co 4852  df-dm 4853  df-rn 4854  df-res 4855  df-ima 4856  df-iota 5384  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-ov 6097  df-oprab 6098  df-mpt2 6099  df-1st 6580  df-2nd 6581  df-map 7219  df-ixp 7267  df-func 14771  df-full 14817  df-fth 14818
This theorem is referenced by:  catcisolem  14977
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