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Theorem coffth 14092
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 3525 . . . 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 3310 . . 3  |-  ( ph  ->  F  e.  ( C Full 
D ) )
4 inss1 3525 . . . 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 3310 . . 3  |-  ( ph  ->  G  e.  ( D Full 
E ) )
73, 6cofull 14090 . 2  |-  ( ph  ->  ( G  o.func  F )  e.  ( C Full  E ) )
8 inss2 3526 . . . 4  |-  ( ( C Full  D )  i^i  ( C Faith  D ) )  C_  ( C Faith  D )
98, 2sseldi 3310 . . 3  |-  ( ph  ->  F  e.  ( C Faith 
D ) )
10 inss2 3526 . . . 4  |-  ( ( D Full  E )  i^i  ( D Faith  E ) )  C_  ( D Faith  E )
1110, 5sseldi 3310 . . 3  |-  ( ph  ->  G  e.  ( D Faith 
E ) )
129, 11cofth 14091 . 2  |-  ( ph  ->  ( G  o.func  F )  e.  ( C Faith  E ) )
13 elin 3494 . 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 646 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 1721    i^i cin 3283  (class class class)co 6044    o.func ccofu 14012   Full cful 14058   Faith cfth 14059
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-rep 4284  ax-sep 4294  ax-nul 4302  ax-pow 4341  ax-pr 4367  ax-un 4664
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-ral 2675  df-rex 2676  df-reu 2677  df-rmo 2678  df-rab 2679  df-v 2922  df-sbc 3126  df-csb 3216  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-nul 3593  df-if 3704  df-pw 3765  df-sn 3784  df-pr 3785  df-op 3787  df-uni 3980  df-iun 4059  df-br 4177  df-opab 4231  df-mpt 4232  df-id 4462  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-rn 4852  df-res 4853  df-ima 4854  df-iota 5381  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-ov 6047  df-oprab 6048  df-mpt2 6049  df-1st 6312  df-2nd 6313  df-riota 6512  df-map 6983  df-ixp 7027  df-cat 13852  df-cid 13853  df-func 14014  df-cofu 14016  df-full 14060  df-fth 14061
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