Theorem coffth 17928

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 4196	. . 3 ⊢ (𝜑 → 𝐹 ∈ (𝐶 Full 𝐷))
3		coffth.g	. . . 4 ⊢ (𝜑 → 𝐺 ∈ ((𝐷 Full 𝐸) ∩ (𝐷 Faith 𝐸)))
4	3	elin1d 4196	. . 3 ⊢ (𝜑 → 𝐺 ∈ (𝐷 Full 𝐸))
5	2, 4	cofull 17926	. 2 ⊢ (𝜑 → (𝐺 ∘func 𝐹) ∈ (𝐶 Full 𝐸))
6	1	elin2d 4197	. . 3 ⊢ (𝜑 → 𝐹 ∈ (𝐶 Faith 𝐷))
7	3	elin2d 4197	. . 3 ⊢ (𝜑 → 𝐺 ∈ (𝐷 Faith 𝐸))
8	6, 7	cofth 17927	. 2 ⊢ (𝜑 → (𝐺 ∘func 𝐹) ∈ (𝐶 Faith 𝐸))
9	5, 8	elind 4192	1 ⊢ (𝜑 → (𝐺 ∘func 𝐹) ∈ ((𝐶 Full 𝐸) ∩ (𝐶 Faith 𝐸)))

Syntax hints:   → wi 4   ∈ wcel 2098   ∩ cin 3943  (class class class)co 7419   ∘_func ccofu 17845   Full cful 17894   Faith cfth 17895

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-un 7741

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-rmo 3363  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-iun 4999  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fo 6555  df-f1o 6556  df-fv 6557  df-riota 7375  df-ov 7422  df-oprab 7423  df-mpo 7424  df-1st 7994  df-2nd 7995  df-map 8847  df-ixp 8917  df-cat 17651  df-cid 17652  df-func 17847  df-cofu 17849  df-full 17896  df-fth 17897

This theorem is referenced by: (None)