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Theorem coffth 16419
 Description: The composition of two fully faithful functors is fully faithful. (Contributed by Mario Carneiro, 28-Jan-2017.)
Hypotheses
Ref Expression
coffth.f (𝜑𝐹 ∈ ((𝐶 Full 𝐷) ∩ (𝐶 Faith 𝐷)))
coffth.g (𝜑𝐺 ∈ ((𝐷 Full 𝐸) ∩ (𝐷 Faith 𝐸)))
Assertion
Ref Expression
coffth (𝜑 → (𝐺func 𝐹) ∈ ((𝐶 Full 𝐸) ∩ (𝐶 Faith 𝐸)))

Proof of Theorem coffth
StepHypRef Expression
1 inss1 3795 . . . 4 ((𝐶 Full 𝐷) ∩ (𝐶 Faith 𝐷)) ⊆ (𝐶 Full 𝐷)
2 coffth.f . . . 4 (𝜑𝐹 ∈ ((𝐶 Full 𝐷) ∩ (𝐶 Faith 𝐷)))
31, 2sseldi 3566 . . 3 (𝜑𝐹 ∈ (𝐶 Full 𝐷))
4 inss1 3795 . . . 4 ((𝐷 Full 𝐸) ∩ (𝐷 Faith 𝐸)) ⊆ (𝐷 Full 𝐸)
5 coffth.g . . . 4 (𝜑𝐺 ∈ ((𝐷 Full 𝐸) ∩ (𝐷 Faith 𝐸)))
64, 5sseldi 3566 . . 3 (𝜑𝐺 ∈ (𝐷 Full 𝐸))
73, 6cofull 16417 . 2 (𝜑 → (𝐺func 𝐹) ∈ (𝐶 Full 𝐸))
8 inss2 3796 . . . 4 ((𝐶 Full 𝐷) ∩ (𝐶 Faith 𝐷)) ⊆ (𝐶 Faith 𝐷)
98, 2sseldi 3566 . . 3 (𝜑𝐹 ∈ (𝐶 Faith 𝐷))
10 inss2 3796 . . . 4 ((𝐷 Full 𝐸) ∩ (𝐷 Faith 𝐸)) ⊆ (𝐷 Faith 𝐸)
1110, 5sseldi 3566 . . 3 (𝜑𝐺 ∈ (𝐷 Faith 𝐸))
129, 11cofth 16418 . 2 (𝜑 → (𝐺func 𝐹) ∈ (𝐶 Faith 𝐸))
137, 12elind 3760 1 (𝜑 → (𝐺func 𝐹) ∈ ((𝐶 Full 𝐸) ∩ (𝐶 Faith 𝐸)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∈ wcel 1977   ∩ cin 3539  (class class class)co 6549   ∘func ccofu 16339   Full cful 16385   Faith cfth 16386 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-1st 7059  df-2nd 7060  df-map 7746  df-ixp 7795  df-cat 16152  df-cid 16153  df-func 16341  df-cofu 16343  df-full 16387  df-fth 16388 This theorem is referenced by: (None)
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