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 4157	. . 3 ⊢ (𝜑 → 𝐹 ∈ (𝐶 Full 𝐷))
3		coffth.g	. . . 4 ⊢ (𝜑 → 𝐺 ∈ ((𝐷 Full 𝐸) ∩ (𝐷 Faith 𝐸)))
4	3	elin1d 4157	. . 3 ⊢ (𝜑 → 𝐺 ∈ (𝐷 Full 𝐸))
5	2, 4	cofull 17988	. 2 ⊢ (𝜑 → (𝐺 ∘func 𝐹) ∈ (𝐶 Full 𝐸))
6	1	elin2d 4158	. . 3 ⊢ (𝜑 → 𝐹 ∈ (𝐶 Faith 𝐷))
7	3	elin2d 4158	. . 3 ⊢ (𝜑 → 𝐺 ∈ (𝐷 Faith 𝐸))
8	6, 7	cofth 17989	. 2 ⊢ (𝜑 → (𝐺 ∘func 𝐹) ∈ (𝐶 Faith 𝐸))
9	5, 8	elind 4153	1 ⊢ (𝜑 → (𝐺 ∘func 𝐹) ∈ ((𝐶 Full 𝐸) ∩ (𝐶 Faith 𝐸)))

Syntax hints:   → wi 4   ∈ wcel 2143   ∩ cin 3904  (class class class)co 7410   ∘_func ccofu 17908   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 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5238  ax-sep 5257  ax-nul 5269  ax-pow 5336  ax-pr 5404  ax-un 7732

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-iun 4958  df-br 5110  df-opab 5174  df-mpt 5193  df-id 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7367  df-ov 7413  df-oprab 7414  df-mpo 7415  df-1st 7982  df-2nd 7983  df-map 8822  df-ixp 8892  df-cat 17719  df-cid 17720  df-func 17910  df-cofu 17912  df-full 17958  df-fth 17959

This theorem is referenced by: (None)