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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 17413 | . . . . 5 ⊢ (𝐶 Full 𝐷) ⊆ (𝐶 Func 𝐷) | |
5 | 4 | ssbri 5098 | . . . 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 17381 | . 2 ⊢ (𝜑 → 𝐹(𝑂 Func 𝑃)tpos 𝐺) |
8 | eqid 2737 | . . . . . 6 ⊢ (Base‘𝐶) = (Base‘𝐶) | |
9 | eqid 2737 | . . . . . 6 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷) | |
10 | eqid 2737 | . . . . . 6 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶) | |
11 | 3 | adantr 484 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝐹(𝐶 Full 𝐷)𝐺) |
12 | simprr 773 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝑦 ∈ (Base‘𝐶)) | |
13 | simprl 771 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝑥 ∈ (Base‘𝐶)) | |
14 | 8, 9, 10, 11, 12, 13 | fullfo 17419 | . . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (𝑦𝐺𝑥):(𝑦(Hom ‘𝐶)𝑥)–onto→((𝐹‘𝑦)(Hom ‘𝐷)(𝐹‘𝑥))) |
15 | forn 6636 | . . . . 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 7983 | . . . . 5 ⊢ (𝑥tpos 𝐺𝑦) = (𝑦𝐺𝑥) | |
18 | 17 | rneqi 5806 | . . . 4 ⊢ ran (𝑥tpos 𝐺𝑦) = ran (𝑦𝐺𝑥) |
19 | 9, 2 | oppchom 17219 | . . . 4 ⊢ ((𝐹‘𝑥)(Hom ‘𝑃)(𝐹‘𝑦)) = ((𝐹‘𝑦)(Hom ‘𝐷)(𝐹‘𝑥)) |
20 | 16, 18, 19 | 3eqtr4g 2803 | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → ran (𝑥tpos 𝐺𝑦) = ((𝐹‘𝑥)(Hom ‘𝑃)(𝐹‘𝑦))) |
21 | 20 | ralrimivva 3112 | . 2 ⊢ (𝜑 → ∀𝑥 ∈ (Base‘𝐶)∀𝑦 ∈ (Base‘𝐶)ran (𝑥tpos 𝐺𝑦) = ((𝐹‘𝑥)(Hom ‘𝑃)(𝐹‘𝑦))) |
22 | 1, 8 | oppcbas 17222 | . . 3 ⊢ (Base‘𝐶) = (Base‘𝑂) |
23 | eqid 2737 | . . 3 ⊢ (Hom ‘𝑃) = (Hom ‘𝑃) | |
24 | 22, 23 | isfull 17417 | . 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 586 | 1 ⊢ (𝜑 → 𝐹(𝑂 Full 𝑃)tpos 𝐺) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1543 ∈ wcel 2110 ∀wral 3061 class class class wbr 5053 ran crn 5552 –onto→wfo 6378 ‘cfv 6380 (class class class)co 7213 tpos ctpos 7967 Basecbs 16760 Hom chom 16813 oppCatcoppc 17214 Func cfunc 17360 Full cful 17409 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-rep 5179 ax-sep 5192 ax-nul 5199 ax-pow 5258 ax-pr 5322 ax-un 7523 ax-cnex 10785 ax-resscn 10786 ax-1cn 10787 ax-icn 10788 ax-addcl 10789 ax-addrcl 10790 ax-mulcl 10791 ax-mulrcl 10792 ax-mulcom 10793 ax-addass 10794 ax-mulass 10795 ax-distr 10796 ax-i2m1 10797 ax-1ne0 10798 ax-1rid 10799 ax-rnegex 10800 ax-rrecex 10801 ax-cnre 10802 ax-pre-lttri 10803 ax-pre-lttrn 10804 ax-pre-ltadd 10805 ax-pre-mulgt0 10806 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3410 df-sbc 3695 df-csb 3812 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-pss 3885 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-tp 4546 df-op 4548 df-uni 4820 df-iun 4906 df-br 5054 df-opab 5116 df-mpt 5136 df-tr 5162 df-id 5455 df-eprel 5460 df-po 5468 df-so 5469 df-fr 5509 df-we 5511 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-pred 6160 df-ord 6216 df-on 6217 df-lim 6218 df-suc 6219 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-f1 6385 df-fo 6386 df-f1o 6387 df-fv 6388 df-riota 7170 df-ov 7216 df-oprab 7217 df-mpo 7218 df-om 7645 df-1st 7761 df-2nd 7762 df-tpos 7968 df-wrecs 8047 df-recs 8108 df-rdg 8146 df-er 8391 df-map 8510 df-ixp 8579 df-en 8627 df-dom 8628 df-sdom 8629 df-pnf 10869 df-mnf 10870 df-xr 10871 df-ltxr 10872 df-le 10873 df-sub 11064 df-neg 11065 df-nn 11831 df-2 11893 df-3 11894 df-4 11895 df-5 11896 df-6 11897 df-7 11898 df-8 11899 df-9 11900 df-n0 12091 df-z 12177 df-dec 12294 df-sets 16717 df-slot 16735 df-ndx 16745 df-base 16761 df-hom 16826 df-cco 16827 df-cat 17171 df-cid 17172 df-oppc 17215 df-func 17364 df-full 17411 |
This theorem is referenced by: ffthoppc 17431 |
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