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Theorem coffth 17983
Description: The composition of two fully faithful functors is fully faithful. (Contributed by Mario Carneiro, 28-Jan-2017.)
Hypotheses
Ref Expression
coffth.f (𝜑𝐹 ∈ ((𝐶 Full 𝐷) ∩ (𝐶 Faith 𝐷)))
coffth.g (𝜑𝐺 ∈ ((𝐷 Full 𝐸) ∩ (𝐷 Faith 𝐸)))
Assertion
Ref Expression
coffth (𝜑 → (𝐺func 𝐹) ∈ ((𝐶 Full 𝐸) ∩ (𝐶 Faith 𝐸)))

Proof of Theorem coffth
StepHypRef Expression
1 coffth.f . . . 4 (𝜑𝐹 ∈ ((𝐶 Full 𝐷) ∩ (𝐶 Faith 𝐷)))
21elin1d 4204 . . 3 (𝜑𝐹 ∈ (𝐶 Full 𝐷))
3 coffth.g . . . 4 (𝜑𝐺 ∈ ((𝐷 Full 𝐸) ∩ (𝐷 Faith 𝐸)))
43elin1d 4204 . . 3 (𝜑𝐺 ∈ (𝐷 Full 𝐸))
52, 4cofull 17981 . 2 (𝜑 → (𝐺func 𝐹) ∈ (𝐶 Full 𝐸))
61elin2d 4205 . . 3 (𝜑𝐹 ∈ (𝐶 Faith 𝐷))
73elin2d 4205 . . 3 (𝜑𝐺 ∈ (𝐷 Faith 𝐸))
86, 7cofth 17982 . 2 (𝜑 → (𝐺func 𝐹) ∈ (𝐶 Faith 𝐸))
95, 8elind 4200 1 (𝜑 → (𝐺func 𝐹) ∈ ((𝐶 Full 𝐸) ∩ (𝐶 Faith 𝐸)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2108  cin 3950  (class class class)co 7431  func ccofu 17901   Full cful 17949   Faith cfth 17950
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8014  df-2nd 8015  df-map 8868  df-ixp 8938  df-cat 17711  df-cid 17712  df-func 17903  df-cofu 17905  df-full 17951  df-fth 17952
This theorem is referenced by: (None)
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