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Theorem ffthf1o 14812
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 4329 . . . . 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 460 . . 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 14810 . 2  |-  ( ph  ->  ( X G Y ) : ( X H Y ) -1-1-> ( ( F `  X
) J ( F `
 Y ) ) )
116simpld 456 . . 3  |-  ( ph  ->  F ( C Full  D
) G )
121, 3, 2, 11, 8, 9fullfo 14805 . 2  |-  ( ph  ->  ( X G Y ) : ( X H Y ) -onto-> ( ( F `  X
) J ( F `
 Y ) ) )
13 df-f1o 5413 . 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 657 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 1362    e. wcel 1755    i^i cin 3315   class class class wbr 4280   -1-1->wf1 5403   -onto->wfo 5404   -1-1-onto->wf1o 5405   ` cfv 5406  (class class class)co 6080   Basecbs 14157   Hom chom 14232   Full cful 14795   Faith cfth 14796
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-8 1757  ax-9 1759  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414  ax-rep 4391  ax-sep 4401  ax-nul 4409  ax-pow 4458  ax-pr 4519  ax-un 6361
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 960  df-tru 1365  df-ex 1590  df-nf 1593  df-sb 1700  df-eu 2258  df-mo 2259  df-clab 2420  df-cleq 2426  df-clel 2429  df-nfc 2558  df-ne 2598  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2964  df-sbc 3176  df-csb 3277  df-dif 3319  df-un 3321  df-in 3323  df-ss 3330  df-nul 3626  df-if 3780  df-pw 3850  df-sn 3866  df-pr 3868  df-op 3872  df-uni 4080  df-iun 4161  df-br 4281  df-opab 4339  df-mpt 4340  df-id 4623  df-xp 4833  df-rel 4834  df-cnv 4835  df-co 4836  df-dm 4837  df-rn 4838  df-res 4839  df-ima 4840  df-iota 5369  df-fun 5408  df-fn 5409  df-f 5410  df-f1 5411  df-fo 5412  df-f1o 5413  df-fv 5414  df-ov 6083  df-oprab 6084  df-mpt2 6085  df-1st 6566  df-2nd 6567  df-map 7204  df-ixp 7252  df-func 14751  df-full 14797  df-fth 14798
This theorem is referenced by:  catcisolem  14957
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