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Mirrors > Home > MPE Home > Th. List > ffthoppc | Structured version Visualization version GIF version |
Description: The opposite functor of a fully faithful functor is also full and faithful. (Contributed by Mario Carneiro, 27-Jan-2017.) |
Ref | Expression |
---|---|
fulloppc.o | ⊢ 𝑂 = (oppCat‘𝐶) |
fulloppc.p | ⊢ 𝑃 = (oppCat‘𝐷) |
ffthoppc.f | ⊢ (𝜑 → 𝐹((𝐶 Full 𝐷) ∩ (𝐶 Faith 𝐷))𝐺) |
Ref | Expression |
---|---|
ffthoppc | ⊢ (𝜑 → 𝐹((𝑂 Full 𝑃) ∩ (𝑂 Faith 𝑃))tpos 𝐺) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fulloppc.o | . . 3 ⊢ 𝑂 = (oppCat‘𝐶) | |
2 | fulloppc.p | . . 3 ⊢ 𝑃 = (oppCat‘𝐷) | |
3 | ffthoppc.f | . . . . 5 ⊢ (𝜑 → 𝐹((𝐶 Full 𝐷) ∩ (𝐶 Faith 𝐷))𝐺) | |
4 | brin 4938 | . . . . 5 ⊢ (𝐹((𝐶 Full 𝐷) ∩ (𝐶 Faith 𝐷))𝐺 ↔ (𝐹(𝐶 Full 𝐷)𝐺 ∧ 𝐹(𝐶 Faith 𝐷)𝐺)) | |
5 | 3, 4 | sylib 210 | . . . 4 ⊢ (𝜑 → (𝐹(𝐶 Full 𝐷)𝐺 ∧ 𝐹(𝐶 Faith 𝐷)𝐺)) |
6 | 5 | simpld 490 | . . 3 ⊢ (𝜑 → 𝐹(𝐶 Full 𝐷)𝐺) |
7 | 1, 2, 6 | fulloppc 16967 | . 2 ⊢ (𝜑 → 𝐹(𝑂 Full 𝑃)tpos 𝐺) |
8 | 5 | simprd 491 | . . 3 ⊢ (𝜑 → 𝐹(𝐶 Faith 𝐷)𝐺) |
9 | 1, 2, 8 | fthoppc 16968 | . 2 ⊢ (𝜑 → 𝐹(𝑂 Faith 𝑃)tpos 𝐺) |
10 | brin 4938 | . 2 ⊢ (𝐹((𝑂 Full 𝑃) ∩ (𝑂 Faith 𝑃))tpos 𝐺 ↔ (𝐹(𝑂 Full 𝑃)tpos 𝐺 ∧ 𝐹(𝑂 Faith 𝑃)tpos 𝐺)) | |
11 | 7, 9, 10 | sylanbrc 578 | 1 ⊢ (𝜑 → 𝐹((𝑂 Full 𝑃) ∩ (𝑂 Faith 𝑃))tpos 𝐺) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 = wceq 1601 ∩ cin 3790 class class class wbr 4886 ‘cfv 6135 (class class class)co 6922 tpos ctpos 7633 oppCatcoppc 16756 Full cful 16947 Faith cfth 16948 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2054 ax-8 2108 ax-9 2115 ax-10 2134 ax-11 2149 ax-12 2162 ax-13 2333 ax-ext 2753 ax-rep 5006 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 ax-cnex 10328 ax-resscn 10329 ax-1cn 10330 ax-icn 10331 ax-addcl 10332 ax-addrcl 10333 ax-mulcl 10334 ax-mulrcl 10335 ax-mulcom 10336 ax-addass 10337 ax-mulass 10338 ax-distr 10339 ax-i2m1 10340 ax-1ne0 10341 ax-1rid 10342 ax-rnegex 10343 ax-rrecex 10344 ax-cnre 10345 ax-pre-lttri 10346 ax-pre-lttrn 10347 ax-pre-ltadd 10348 ax-pre-mulgt0 10349 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2550 df-eu 2586 df-clab 2763 df-cleq 2769 df-clel 2773 df-nfc 2920 df-ne 2969 df-nel 3075 df-ral 3094 df-rex 3095 df-reu 3096 df-rmo 3097 df-rab 3098 df-v 3399 df-sbc 3652 df-csb 3751 df-dif 3794 df-un 3796 df-in 3798 df-ss 3805 df-pss 3807 df-nul 4141 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-tp 4402 df-op 4404 df-uni 4672 df-iun 4755 df-br 4887 df-opab 4949 df-mpt 4966 df-tr 4988 df-id 5261 df-eprel 5266 df-po 5274 df-so 5275 df-fr 5314 df-we 5316 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-pred 5933 df-ord 5979 df-on 5980 df-lim 5981 df-suc 5982 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-riota 6883 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-om 7344 df-1st 7445 df-2nd 7446 df-tpos 7634 df-wrecs 7689 df-recs 7751 df-rdg 7789 df-er 8026 df-map 8142 df-ixp 8195 df-en 8242 df-dom 8243 df-sdom 8244 df-pnf 10413 df-mnf 10414 df-xr 10415 df-ltxr 10416 df-le 10417 df-sub 10608 df-neg 10609 df-nn 11375 df-2 11438 df-3 11439 df-4 11440 df-5 11441 df-6 11442 df-7 11443 df-8 11444 df-9 11445 df-n0 11643 df-z 11729 df-dec 11846 df-ndx 16258 df-slot 16259 df-base 16261 df-sets 16262 df-hom 16362 df-cco 16363 df-cat 16714 df-cid 16715 df-oppc 16757 df-func 16903 df-full 16949 df-fth 16950 |
This theorem is referenced by: (None) |
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