Theorem coffth 17962

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 4154	. . 3 ⊢ (𝜑 → 𝐹 ∈ (𝐶 Full 𝐷))
3		coffth.g	. . . 4 ⊢ (𝜑 → 𝐺 ∈ ((𝐷 Full 𝐸) ∩ (𝐷 Faith 𝐸)))
4	3	elin1d 4154	. . 3 ⊢ (𝜑 → 𝐺 ∈ (𝐷 Full 𝐸))
5	2, 4	cofull 17960	. 2 ⊢ (𝜑 → (𝐺 ∘func 𝐹) ∈ (𝐶 Full 𝐸))
6	1	elin2d 4155	. . 3 ⊢ (𝜑 → 𝐹 ∈ (𝐶 Faith 𝐷))
7	3	elin2d 4155	. . 3 ⊢ (𝜑 → 𝐺 ∈ (𝐷 Faith 𝐸))
8	6, 7	cofth 17961	. 2 ⊢ (𝜑 → (𝐺 ∘func 𝐹) ∈ (𝐶 Faith 𝐸))
9	5, 8	elind 4150	1 ⊢ (𝜑 → (𝐺 ∘func 𝐹) ∈ ((𝐶 Full 𝐸) ∩ (𝐶 Faith 𝐸)))

Syntax hints:   → wi 4   ∈ wcel 2141   ∩ cin 3901  (class class class)co 7391   ∘_func ccofu 17880   Full cful 17928   Faith cfth 17929

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5224  ax-sep 5243  ax-nul 5253  ax-pow 5319  ax-pr 5387  ax-un 7713

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-iun 4948  df-br 5098  df-opab 5160  df-mpt 5179  df-id 5538  df-xp 5649  df-rel 5650  df-cnv 5651  df-co 5652  df-dm 5653  df-rn 5654  df-res 5655  df-ima 5656  df-iota 6472  df-fun 6518  df-fn 6519  df-f 6520  df-f1 6521  df-fo 6522  df-f1o 6523  df-fv 6524  df-riota 7348  df-ov 7394  df-oprab 7395  df-mpo 7396  df-1st 7965  df-2nd 7966  df-map 8804  df-ixp 8874  df-cat 17691  df-cid 17692  df-func 17882  df-cofu 17884  df-full 17930  df-fth 17931

This theorem is referenced by: (None)