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Theorem coffth 14867
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 3591 . . . 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 3375 . . 3  |-  ( ph  ->  F  e.  ( C Full 
D ) )
4 inss1 3591 . . . 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 3375 . . 3  |-  ( ph  ->  G  e.  ( D Full 
E ) )
73, 6cofull 14865 . 2  |-  ( ph  ->  ( G  o.func  F )  e.  ( C Full  E ) )
8 inss2 3592 . . . 4  |-  ( ( C Full  D )  i^i  ( C Faith  D ) )  C_  ( C Faith  D )
98, 2sseldi 3375 . . 3  |-  ( ph  ->  F  e.  ( C Faith 
D ) )
10 inss2 3592 . . . 4  |-  ( ( D Full  E )  i^i  ( D Faith  E ) )  C_  ( D Faith  E )
1110, 5sseldi 3375 . . 3  |-  ( ph  ->  G  e.  ( D Faith 
E ) )
129, 11cofth 14866 . 2  |-  ( ph  ->  ( G  o.func  F )  e.  ( C Faith  E ) )
137, 12elind 3561 1  |-  ( ph  ->  ( G  o.func  F )  e.  ( ( C Full  E
)  i^i  ( C Faith  E ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    e. wcel 1756    i^i cin 3348  (class class class)co 6112    o.func ccofu 14787   Full cful 14833   Faith cfth 14834
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 4424  ax-sep 4434  ax-nul 4442  ax-pow 4491  ax-pr 4552  ax-un 6393
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 2577  df-ne 2622  df-ral 2741  df-rex 2742  df-reu 2743  df-rmo 2744  df-rab 2745  df-v 2995  df-sbc 3208  df-csb 3310  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-nul 3659  df-if 3813  df-pw 3883  df-sn 3899  df-pr 3901  df-op 3905  df-uni 4113  df-iun 4194  df-br 4314  df-opab 4372  df-mpt 4373  df-id 4657  df-xp 4867  df-rel 4868  df-cnv 4869  df-co 4870  df-dm 4871  df-rn 4872  df-res 4873  df-ima 4874  df-iota 5402  df-fun 5441  df-fn 5442  df-f 5443  df-f1 5444  df-fo 5445  df-f1o 5446  df-fv 5447  df-riota 6073  df-ov 6115  df-oprab 6116  df-mpt2 6117  df-1st 6598  df-2nd 6599  df-map 7237  df-ixp 7285  df-cat 14627  df-cid 14628  df-func 14789  df-cofu 14791  df-full 14835  df-fth 14836
This theorem is referenced by: (None)
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