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Mirrors > Home > MPE Home > Th. List > coffth | Structured version Visualization version GIF version |
Description: The composition of two fully faithful functors is fully faithful. (Contributed by Mario Carneiro, 28-Jan-2017.) |
Ref | Expression |
---|---|
coffth.f | ⊢ (𝜑 → 𝐹 ∈ ((𝐶 Full 𝐷) ∩ (𝐶 Faith 𝐷))) |
coffth.g | ⊢ (𝜑 → 𝐺 ∈ ((𝐷 Full 𝐸) ∩ (𝐷 Faith 𝐸))) |
Ref | Expression |
---|---|
coffth | ⊢ (𝜑 → (𝐺 ∘func 𝐹) ∈ ((𝐶 Full 𝐸) ∩ (𝐶 Faith 𝐸))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | coffth.f | . . . 4 ⊢ (𝜑 → 𝐹 ∈ ((𝐶 Full 𝐷) ∩ (𝐶 Faith 𝐷))) | |
2 | 1 | elin1d 4159 | . . 3 ⊢ (𝜑 → 𝐹 ∈ (𝐶 Full 𝐷)) |
3 | coffth.g | . . . 4 ⊢ (𝜑 → 𝐺 ∈ ((𝐷 Full 𝐸) ∩ (𝐷 Faith 𝐸))) | |
4 | 3 | elin1d 4159 | . . 3 ⊢ (𝜑 → 𝐺 ∈ (𝐷 Full 𝐸)) |
5 | 2, 4 | cofull 17826 | . 2 ⊢ (𝜑 → (𝐺 ∘func 𝐹) ∈ (𝐶 Full 𝐸)) |
6 | 1 | elin2d 4160 | . . 3 ⊢ (𝜑 → 𝐹 ∈ (𝐶 Faith 𝐷)) |
7 | 3 | elin2d 4160 | . . 3 ⊢ (𝜑 → 𝐺 ∈ (𝐷 Faith 𝐸)) |
8 | 6, 7 | cofth 17827 | . 2 ⊢ (𝜑 → (𝐺 ∘func 𝐹) ∈ (𝐶 Faith 𝐸)) |
9 | 5, 8 | elind 4155 | 1 ⊢ (𝜑 → (𝐺 ∘func 𝐹) ∈ ((𝐶 Full 𝐸) ∩ (𝐶 Faith 𝐸))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2107 ∩ cin 3910 (class class class)co 7358 ∘func ccofu 17747 Full cful 17794 Faith cfth 17795 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5243 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-id 5532 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-1st 7922 df-2nd 7923 df-map 8770 df-ixp 8839 df-cat 17553 df-cid 17554 df-func 17749 df-cofu 17751 df-full 17796 df-fth 17797 |
This theorem is referenced by: (None) |
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