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Theorem coffth 13810
Description: The composition of two fully faithful functors is fully faithful. (Contributed by Mario Carneiro, 28-Jan-2017.)
Hypotheses
Ref Expression
coffth.f  |-  ( ph  ->  F  e.  ( ( C Full  D )  i^i  ( C Faith  D ) ) )
coffth.g  |-  ( ph  ->  G  e.  ( ( D Full  E )  i^i  ( D Faith  E ) ) )
Assertion
Ref Expression
coffth  |-  ( ph  ->  ( G  o.func  F )  e.  ( ( C Full  E
)  i^i  ( C Faith  E ) ) )

Proof of Theorem coffth
StepHypRef Expression
1 inss1 3389 . . . 4  |-  ( ( C Full  D )  i^i  ( C Faith  D ) )  C_  ( C Full  D )
2 coffth.f . . . 4  |-  ( ph  ->  F  e.  ( ( C Full  D )  i^i  ( C Faith  D ) ) )
31, 2sseldi 3178 . . 3  |-  ( ph  ->  F  e.  ( C Full 
D ) )
4 inss1 3389 . . . 4  |-  ( ( D Full  E )  i^i  ( D Faith  E ) )  C_  ( D Full  E )
5 coffth.g . . . 4  |-  ( ph  ->  G  e.  ( ( D Full  E )  i^i  ( D Faith  E ) ) )
64, 5sseldi 3178 . . 3  |-  ( ph  ->  G  e.  ( D Full 
E ) )
73, 6cofull 13808 . 2  |-  ( ph  ->  ( G  o.func  F )  e.  ( C Full  E ) )
8 inss2 3390 . . . 4  |-  ( ( C Full  D )  i^i  ( C Faith  D ) )  C_  ( C Faith  D )
98, 2sseldi 3178 . . 3  |-  ( ph  ->  F  e.  ( C Faith 
D ) )
10 inss2 3390 . . . 4  |-  ( ( D Full  E )  i^i  ( D Faith  E ) )  C_  ( D Faith  E )
1110, 5sseldi 3178 . . 3  |-  ( ph  ->  G  e.  ( D Faith 
E ) )
129, 11cofth 13809 . 2  |-  ( ph  ->  ( G  o.func  F )  e.  ( C Faith  E ) )
13 elin 3358 . 2  |-  ( ( G  o.func 
F )  e.  ( ( C Full  E )  i^i  ( C Faith  E
) )  <->  ( ( G  o.func 
F )  e.  ( C Full  E )  /\  ( G  o.func 
F )  e.  ( C Faith  E ) ) )
147, 12, 13sylanbrc 645 1  |-  ( ph  ->  ( G  o.func  F )  e.  ( ( C Full  E
)  i^i  ( C Faith  E ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1684    i^i cin 3151  (class class class)co 5858    o.func ccofu 13730   Full cful 13776   Faith cfth 13777
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-rmo 2551  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-riota 6304  df-map 6774  df-ixp 6818  df-cat 13570  df-cid 13571  df-func 13732  df-cofu 13734  df-full 13778  df-fth 13779
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