Theorem coffth 17864

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 17862	. 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 17863	. 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 2109   ∩ cin 3904  (class class class)co 7353   ∘_func ccofu 17782   Full cful 17830   Faith cfth 17831

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5221  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7675

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rmo 3345  df-reu 3346  df-rab 3397  df-v 3440  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-iun 4946  df-br 5096  df-opab 5158  df-mpt 5177  df-id 5518  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-riota 7310  df-ov 7356  df-oprab 7357  df-mpo 7358  df-1st 7931  df-2nd 7932  df-map 8762  df-ixp 8832  df-cat 17593  df-cid 17594  df-func 17784  df-cofu 17786  df-full 17832  df-fth 17833

This theorem is referenced by: (None)