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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 17178 | . . . . 5 ⊢ (𝐶 Full 𝐷) ⊆ (𝐶 Func 𝐷) | |
5 | 4 | ssbri 5113 | . . . 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 17147 | . 2 ⊢ (𝜑 → 𝐹(𝑂 Func 𝑃)tpos 𝐺) |
8 | eqid 2823 | . . . . . 6 ⊢ (Base‘𝐶) = (Base‘𝐶) | |
9 | eqid 2823 | . . . . . 6 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷) | |
10 | eqid 2823 | . . . . . 6 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶) | |
11 | 3 | adantr 483 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝐹(𝐶 Full 𝐷)𝐺) |
12 | simprr 771 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝑦 ∈ (Base‘𝐶)) | |
13 | simprl 769 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝑥 ∈ (Base‘𝐶)) | |
14 | 8, 9, 10, 11, 12, 13 | fullfo 17184 | . . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (𝑦𝐺𝑥):(𝑦(Hom ‘𝐶)𝑥)–onto→((𝐹‘𝑦)(Hom ‘𝐷)(𝐹‘𝑥))) |
15 | forn 6595 | . . . . 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 7909 | . . . . 5 ⊢ (𝑥tpos 𝐺𝑦) = (𝑦𝐺𝑥) | |
18 | 17 | rneqi 5809 | . . . 4 ⊢ ran (𝑥tpos 𝐺𝑦) = ran (𝑦𝐺𝑥) |
19 | 9, 2 | oppchom 16987 | . . . 4 ⊢ ((𝐹‘𝑥)(Hom ‘𝑃)(𝐹‘𝑦)) = ((𝐹‘𝑦)(Hom ‘𝐷)(𝐹‘𝑥)) |
20 | 16, 18, 19 | 3eqtr4g 2883 | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → ran (𝑥tpos 𝐺𝑦) = ((𝐹‘𝑥)(Hom ‘𝑃)(𝐹‘𝑦))) |
21 | 20 | ralrimivva 3193 | . 2 ⊢ (𝜑 → ∀𝑥 ∈ (Base‘𝐶)∀𝑦 ∈ (Base‘𝐶)ran (𝑥tpos 𝐺𝑦) = ((𝐹‘𝑥)(Hom ‘𝑃)(𝐹‘𝑦))) |
22 | 1, 8 | oppcbas 16990 | . . 3 ⊢ (Base‘𝐶) = (Base‘𝑂) |
23 | eqid 2823 | . . 3 ⊢ (Hom ‘𝑃) = (Hom ‘𝑃) | |
24 | 22, 23 | isfull 17182 | . 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 585 | 1 ⊢ (𝜑 → 𝐹(𝑂 Full 𝑃)tpos 𝐺) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∀wral 3140 class class class wbr 5068 ran crn 5558 –onto→wfo 6355 ‘cfv 6357 (class class class)co 7158 tpos ctpos 7893 Basecbs 16485 Hom chom 16578 oppCatcoppc 16983 Func cfunc 17126 Full cful 17174 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-tpos 7894 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-er 8291 df-map 8410 df-ixp 8464 df-en 8512 df-dom 8513 df-sdom 8514 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-nn 11641 df-2 11703 df-3 11704 df-4 11705 df-5 11706 df-6 11707 df-7 11708 df-8 11709 df-9 11710 df-n0 11901 df-z 11985 df-dec 12102 df-ndx 16488 df-slot 16489 df-base 16491 df-sets 16492 df-hom 16591 df-cco 16592 df-cat 16941 df-cid 16942 df-oppc 16984 df-func 17130 df-full 17176 |
This theorem is referenced by: ffthoppc 17196 |
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