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Theorem coffth 16365
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 3794 . . . 4 ((𝐶 Full 𝐷) ∩ (𝐶 Faith 𝐷)) ⊆ (𝐶 Full 𝐷)
2 coffth.f . . . 4 (𝜑𝐹 ∈ ((𝐶 Full 𝐷) ∩ (𝐶 Faith 𝐷)))
31, 2sseldi 3565 . . 3 (𝜑𝐹 ∈ (𝐶 Full 𝐷))
4 inss1 3794 . . . 4 ((𝐷 Full 𝐸) ∩ (𝐷 Faith 𝐸)) ⊆ (𝐷 Full 𝐸)
5 coffth.g . . . 4 (𝜑𝐺 ∈ ((𝐷 Full 𝐸) ∩ (𝐷 Faith 𝐸)))
64, 5sseldi 3565 . . 3 (𝜑𝐺 ∈ (𝐷 Full 𝐸))
73, 6cofull 16363 . 2 (𝜑 → (𝐺func 𝐹) ∈ (𝐶 Full 𝐸))
8 inss2 3795 . . . 4 ((𝐶 Full 𝐷) ∩ (𝐶 Faith 𝐷)) ⊆ (𝐶 Faith 𝐷)
98, 2sseldi 3565 . . 3 (𝜑𝐹 ∈ (𝐶 Faith 𝐷))
10 inss2 3795 . . . 4 ((𝐷 Full 𝐸) ∩ (𝐷 Faith 𝐸)) ⊆ (𝐷 Faith 𝐸)
1110, 5sseldi 3565 . . 3 (𝜑𝐺 ∈ (𝐷 Faith 𝐸))
129, 11cofth 16364 . 2 (𝜑 → (𝐺func 𝐹) ∈ (𝐶 Faith 𝐸))
137, 12elind 3759 1 (𝜑 → (𝐺func 𝐹) ∈ ((𝐶 Full 𝐸) ∩ (𝐶 Faith 𝐸)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 1976  cin 3538  (class class class)co 6527  func ccofu 16285   Full cful 16331   Faith cfth 16332
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2232  ax-ext 2589  ax-rep 4693  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6824
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-ral 2900  df-rex 2901  df-reu 2902  df-rmo 2903  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-op 4131  df-uni 4367  df-iun 4451  df-br 4578  df-opab 4638  df-mpt 4639  df-id 4943  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-riota 6489  df-ov 6530  df-oprab 6531  df-mpt2 6532  df-1st 7036  df-2nd 7037  df-map 7723  df-ixp 7772  df-cat 16098  df-cid 16099  df-func 16287  df-cofu 16289  df-full 16333  df-fth 16334
This theorem is referenced by: (None)
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