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Theorem coffth 14164
 Description: The composition of two fully faithful functors is fully faithful. (Contributed by Mario Carneiro, 28-Jan-2017.)
Hypotheses
Ref Expression
coffth.f Full Faith
coffth.g Full Faith
Assertion
Ref Expression
coffth func Full Faith

Proof of Theorem coffth
StepHypRef Expression
1 inss1 3546 . . . 4 Full Faith Full
2 coffth.f . . . 4 Full Faith
31, 2sseldi 3332 . . 3 Full
4 inss1 3546 . . . 4 Full Faith Full
5 coffth.g . . . 4 Full Faith
64, 5sseldi 3332 . . 3 Full
73, 6cofull 14162 . 2 func Full
8 inss2 3547 . . . 4 Full Faith Faith
98, 2sseldi 3332 . . 3 Faith
10 inss2 3547 . . . 4 Full Faith Faith
1110, 5sseldi 3332 . . 3 Faith
129, 11cofth 14163 . 2 func Faith
13 elin 3516 . 2 func Full Faith func Full func Faith
147, 12, 13sylanbrc 647 1 func Full Faith
 Colors of variables: wff set class Syntax hints:   wi 4   wcel 1727   cin 3305  (class class class)co 6110   func ccofu 14084   Full cful 14130   Faith cfth 14131 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-13 1729  ax-14 1731  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423  ax-rep 4345  ax-sep 4355  ax-nul 4363  ax-pow 4406  ax-pr 4432  ax-un 4730 This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2291  df-mo 2292  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2716  df-rex 2717  df-reu 2718  df-rmo 2719  df-rab 2720  df-v 2964  df-sbc 3168  df-csb 3268  df-dif 3309  df-un 3311  df-in 3313  df-ss 3320  df-nul 3614  df-if 3764  df-pw 3825  df-sn 3844  df-pr 3845  df-op 3847  df-uni 4040  df-iun 4119  df-br 4238  df-opab 4292  df-mpt 4293  df-id 4527  df-xp 4913  df-rel 4914  df-cnv 4915  df-co 4916  df-dm 4917  df-rn 4918  df-res 4919  df-ima 4920  df-iota 5447  df-fun 5485  df-fn 5486  df-f 5487  df-f1 5488  df-fo 5489  df-f1o 5490  df-fv 5491  df-ov 6113  df-oprab 6114  df-mpt2 6115  df-1st 6378  df-2nd 6379  df-riota 6578  df-map 7049  df-ixp 7093  df-cat 13924  df-cid 13925  df-func 14086  df-cofu 14088  df-full 14132  df-fth 14133
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