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Mirrors > Home > MPE Home > Th. List > fulloppc | Structured version Visualization version GIF version |
Description: The opposite functor of a full functor is also full. Proposition 3.43(d) in [Adamek] p. 39. (Contributed by Mario Carneiro, 27-Jan-2017.) |
Ref | Expression |
---|---|
fulloppc.o | ⊢ 𝑂 = (oppCat‘𝐶) |
fulloppc.p | ⊢ 𝑃 = (oppCat‘𝐷) |
fulloppc.f | ⊢ (𝜑 → 𝐹(𝐶 Full 𝐷)𝐺) |
Ref | Expression |
---|---|
fulloppc | ⊢ (𝜑 → 𝐹(𝑂 Full 𝑃)tpos 𝐺) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fulloppc.o | . . 3 ⊢ 𝑂 = (oppCat‘𝐶) | |
2 | fulloppc.p | . . 3 ⊢ 𝑃 = (oppCat‘𝐷) | |
3 | fulloppc.f | . . . 4 ⊢ (𝜑 → 𝐹(𝐶 Full 𝐷)𝐺) | |
4 | fullfunc 17612 | . . . . 5 ⊢ (𝐶 Full 𝐷) ⊆ (𝐶 Func 𝐷) | |
5 | 4 | ssbri 5124 | . . . 4 ⊢ (𝐹(𝐶 Full 𝐷)𝐺 → 𝐹(𝐶 Func 𝐷)𝐺) |
6 | 3, 5 | syl 17 | . . 3 ⊢ (𝜑 → 𝐹(𝐶 Func 𝐷)𝐺) |
7 | 1, 2, 6 | funcoppc 17580 | . 2 ⊢ (𝜑 → 𝐹(𝑂 Func 𝑃)tpos 𝐺) |
8 | eqid 2740 | . . . . . 6 ⊢ (Base‘𝐶) = (Base‘𝐶) | |
9 | eqid 2740 | . . . . . 6 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷) | |
10 | eqid 2740 | . . . . . 6 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶) | |
11 | 3 | adantr 481 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝐹(𝐶 Full 𝐷)𝐺) |
12 | simprr 770 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝑦 ∈ (Base‘𝐶)) | |
13 | simprl 768 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝑥 ∈ (Base‘𝐶)) | |
14 | 8, 9, 10, 11, 12, 13 | fullfo 17618 | . . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (𝑦𝐺𝑥):(𝑦(Hom ‘𝐶)𝑥)–onto→((𝐹‘𝑦)(Hom ‘𝐷)(𝐹‘𝑥))) |
15 | forn 6688 | . . . . 5 ⊢ ((𝑦𝐺𝑥):(𝑦(Hom ‘𝐶)𝑥)–onto→((𝐹‘𝑦)(Hom ‘𝐷)(𝐹‘𝑥)) → ran (𝑦𝐺𝑥) = ((𝐹‘𝑦)(Hom ‘𝐷)(𝐹‘𝑥))) | |
16 | 14, 15 | syl 17 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → ran (𝑦𝐺𝑥) = ((𝐹‘𝑦)(Hom ‘𝐷)(𝐹‘𝑥))) |
17 | ovtpos 8042 | . . . . 5 ⊢ (𝑥tpos 𝐺𝑦) = (𝑦𝐺𝑥) | |
18 | 17 | rneqi 5844 | . . . 4 ⊢ ran (𝑥tpos 𝐺𝑦) = ran (𝑦𝐺𝑥) |
19 | 9, 2 | oppchom 17415 | . . . 4 ⊢ ((𝐹‘𝑥)(Hom ‘𝑃)(𝐹‘𝑦)) = ((𝐹‘𝑦)(Hom ‘𝐷)(𝐹‘𝑥)) |
20 | 16, 18, 19 | 3eqtr4g 2805 | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → ran (𝑥tpos 𝐺𝑦) = ((𝐹‘𝑥)(Hom ‘𝑃)(𝐹‘𝑦))) |
21 | 20 | ralrimivva 3117 | . 2 ⊢ (𝜑 → ∀𝑥 ∈ (Base‘𝐶)∀𝑦 ∈ (Base‘𝐶)ran (𝑥tpos 𝐺𝑦) = ((𝐹‘𝑥)(Hom ‘𝑃)(𝐹‘𝑦))) |
22 | 1, 8 | oppcbas 17418 | . . 3 ⊢ (Base‘𝐶) = (Base‘𝑂) |
23 | eqid 2740 | . . 3 ⊢ (Hom ‘𝑃) = (Hom ‘𝑃) | |
24 | 22, 23 | isfull 17616 | . 2 ⊢ (𝐹(𝑂 Full 𝑃)tpos 𝐺 ↔ (𝐹(𝑂 Func 𝑃)tpos 𝐺 ∧ ∀𝑥 ∈ (Base‘𝐶)∀𝑦 ∈ (Base‘𝐶)ran (𝑥tpos 𝐺𝑦) = ((𝐹‘𝑥)(Hom ‘𝑃)(𝐹‘𝑦)))) |
25 | 7, 21, 24 | sylanbrc 583 | 1 ⊢ (𝜑 → 𝐹(𝑂 Full 𝑃)tpos 𝐺) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1542 ∈ wcel 2110 ∀wral 3066 class class class wbr 5079 ran crn 5590 –onto→wfo 6429 ‘cfv 6431 (class class class)co 7269 tpos ctpos 8026 Basecbs 16902 Hom chom 16963 oppCatcoppc 17410 Func cfunc 17559 Full cful 17608 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 ax-rep 5214 ax-sep 5227 ax-nul 5234 ax-pow 5292 ax-pr 5356 ax-un 7580 ax-cnex 10920 ax-resscn 10921 ax-1cn 10922 ax-icn 10923 ax-addcl 10924 ax-addrcl 10925 ax-mulcl 10926 ax-mulrcl 10927 ax-mulcom 10928 ax-addass 10929 ax-mulass 10930 ax-distr 10931 ax-i2m1 10932 ax-1ne0 10933 ax-1rid 10934 ax-rnegex 10935 ax-rrecex 10936 ax-cnre 10937 ax-pre-lttri 10938 ax-pre-lttrn 10939 ax-pre-ltadd 10940 ax-pre-mulgt0 10941 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2072 df-mo 2542 df-eu 2571 df-clab 2718 df-cleq 2732 df-clel 2818 df-nfc 2891 df-ne 2946 df-nel 3052 df-ral 3071 df-rex 3072 df-reu 3073 df-rmo 3074 df-rab 3075 df-v 3433 df-sbc 3721 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-pss 3911 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4846 df-iun 4932 df-br 5080 df-opab 5142 df-mpt 5163 df-tr 5197 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6200 df-ord 6267 df-on 6268 df-lim 6269 df-suc 6270 df-iota 6389 df-fun 6433 df-fn 6434 df-f 6435 df-f1 6436 df-fo 6437 df-f1o 6438 df-fv 6439 df-riota 7226 df-ov 7272 df-oprab 7273 df-mpo 7274 df-om 7702 df-1st 7818 df-2nd 7819 df-tpos 8027 df-frecs 8082 df-wrecs 8113 df-recs 8187 df-rdg 8226 df-er 8473 df-map 8592 df-ixp 8661 df-en 8709 df-dom 8710 df-sdom 8711 df-pnf 11004 df-mnf 11005 df-xr 11006 df-ltxr 11007 df-le 11008 df-sub 11199 df-neg 11200 df-nn 11966 df-2 12028 df-3 12029 df-4 12030 df-5 12031 df-6 12032 df-7 12033 df-8 12034 df-9 12035 df-n0 12226 df-z 12312 df-dec 12429 df-sets 16855 df-slot 16873 df-ndx 16885 df-base 16903 df-hom 16976 df-cco 16977 df-cat 17367 df-cid 17368 df-oppc 17411 df-func 17563 df-full 17610 |
This theorem is referenced by: ffthoppc 17630 |
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