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Theorem coffth 14020
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 3477 . . . 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 3264 . . 3  |-  ( ph  ->  F  e.  ( C Full 
D ) )
4 inss1 3477 . . . 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 3264 . . 3  |-  ( ph  ->  G  e.  ( D Full 
E ) )
73, 6cofull 14018 . 2  |-  ( ph  ->  ( G  o.func  F )  e.  ( C Full  E ) )
8 inss2 3478 . . . 4  |-  ( ( C Full  D )  i^i  ( C Faith  D ) )  C_  ( C Faith  D )
98, 2sseldi 3264 . . 3  |-  ( ph  ->  F  e.  ( C Faith 
D ) )
10 inss2 3478 . . . 4  |-  ( ( D Full  E )  i^i  ( D Faith  E ) )  C_  ( D Faith  E )
1110, 5sseldi 3264 . . 3  |-  ( ph  ->  G  e.  ( D Faith 
E ) )
129, 11cofth 14019 . 2  |-  ( ph  ->  ( G  o.func  F )  e.  ( C Faith  E ) )
13 elin 3446 . 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 1715    i^i cin 3237  (class class class)co 5981    o.func ccofu 13940   Full cful 13986   Faith cfth 13987
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-rep 4233  ax-sep 4243  ax-nul 4251  ax-pow 4290  ax-pr 4316  ax-un 4615
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-ral 2633  df-rex 2634  df-reu 2635  df-rmo 2636  df-rab 2637  df-v 2875  df-sbc 3078  df-csb 3168  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-nul 3544  df-if 3655  df-pw 3716  df-sn 3735  df-pr 3736  df-op 3738  df-uni 3930  df-iun 4009  df-br 4126  df-opab 4180  df-mpt 4181  df-id 4412  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-rn 4803  df-res 4804  df-ima 4805  df-iota 5322  df-fun 5360  df-fn 5361  df-f 5362  df-f1 5363  df-fo 5364  df-f1o 5365  df-fv 5366  df-ov 5984  df-oprab 5985  df-mpt2 5986  df-1st 6249  df-2nd 6250  df-riota 6446  df-map 6917  df-ixp 6961  df-cat 13780  df-cid 13781  df-func 13942  df-cofu 13944  df-full 13988  df-fth 13989
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