Theorem coffth 17990

Description: The composition of two fully faithful functors is fully faithful. (Contributed by Mario Carneiro, 28-Jan-2017.)

Hypotheses

Ref	Expression
coffth.f	⊢ (𝜑 → 𝐹 ∈ ((𝐶 Full 𝐷) ∩ (𝐶 Faith 𝐷)))
coffth.g	⊢ (𝜑 → 𝐺 ∈ ((𝐷 Full 𝐸) ∩ (𝐷 Faith 𝐸)))

Assertion

Ref	Expression
coffth	⊢ (𝜑 → (𝐺 ∘func 𝐹) ∈ ((𝐶 Full 𝐸) ∩ (𝐶 Faith 𝐸)))

Proof of Theorem coffth

Step	Hyp	Ref	Expression
1		coffth.f	. . . 4 ⊢ (𝜑 → 𝐹 ∈ ((𝐶 Full 𝐷) ∩ (𝐶 Faith 𝐷)))
2	1	elin1d 4214	. . 3 ⊢ (𝜑 → 𝐹 ∈ (𝐶 Full 𝐷))
3		coffth.g	. . . 4 ⊢ (𝜑 → 𝐺 ∈ ((𝐷 Full 𝐸) ∩ (𝐷 Faith 𝐸)))
4	3	elin1d 4214	. . 3 ⊢ (𝜑 → 𝐺 ∈ (𝐷 Full 𝐸))
5	2, 4	cofull 17988	. 2 ⊢ (𝜑 → (𝐺 ∘func 𝐹) ∈ (𝐶 Full 𝐸))
6	1	elin2d 4215	. . 3 ⊢ (𝜑 → 𝐹 ∈ (𝐶 Faith 𝐷))
7	3	elin2d 4215	. . 3 ⊢ (𝜑 → 𝐺 ∈ (𝐷 Faith 𝐸))
8	6, 7	cofth 17989	. 2 ⊢ (𝜑 → (𝐺 ∘func 𝐹) ∈ (𝐶 Faith 𝐸))
9	5, 8	elind 4210	1 ⊢ (𝜑 → (𝐺 ∘func 𝐹) ∈ ((𝐶 Full 𝐸) ∩ (𝐶 Faith 𝐸)))

Syntax hints:   → wi 4   ∈ wcel 2106   ∩ cin 3962  (class class class)co 7431   ∘_func ccofu 17907   Full cful 17956   Faith cfth 17957

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8013  df-2nd 8014  df-map 8867  df-ixp 8937  df-cat 17713  df-cid 17714  df-func 17909  df-cofu 17911  df-full 17958  df-fth 17959

This theorem is referenced by: (None)