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Theorem coffth 15351
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 3714 . . . 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 3497 . . 3  |-  ( ph  ->  F  e.  ( C Full 
D ) )
4 inss1 3714 . . . 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 3497 . . 3  |-  ( ph  ->  G  e.  ( D Full 
E ) )
73, 6cofull 15349 . 2  |-  ( ph  ->  ( G  o.func  F )  e.  ( C Full  E ) )
8 inss2 3715 . . . 4  |-  ( ( C Full  D )  i^i  ( C Faith  D ) )  C_  ( C Faith  D )
98, 2sseldi 3497 . . 3  |-  ( ph  ->  F  e.  ( C Faith 
D ) )
10 inss2 3715 . . . 4  |-  ( ( D Full  E )  i^i  ( D Faith  E ) )  C_  ( D Faith  E )
1110, 5sseldi 3497 . . 3  |-  ( ph  ->  G  e.  ( D Faith 
E ) )
129, 11cofth 15350 . 2  |-  ( ph  ->  ( G  o.func  F )  e.  ( C Faith  E ) )
137, 12elind 3684 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 1819    i^i cin 3470  (class class class)co 6296    o.func ccofu 15271   Full cful 15317   Faith cfth 15318
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-1st 6799  df-2nd 6800  df-map 7440  df-ixp 7489  df-cat 15084  df-cid 15085  df-func 15273  df-cofu 15275  df-full 15319  df-fth 15320
This theorem is referenced by: (None)
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