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Mirrors > Home > MPE Home > Th. List > oppchofcl | Structured version Visualization version GIF version |
Description: Closure of the opposite Hom functor. (Contributed by Mario Carneiro, 17-Jan-2017.) |
Ref | Expression |
---|---|
oppchofcl.o | ⊢ 𝑂 = (oppCat‘𝐶) |
oppchofcl.m | ⊢ 𝑀 = (HomF‘𝑂) |
oppchofcl.d | ⊢ 𝐷 = (SetCat‘𝑈) |
oppchofcl.c | ⊢ (𝜑 → 𝐶 ∈ Cat) |
oppchofcl.u | ⊢ (𝜑 → 𝑈 ∈ 𝑉) |
oppchofcl.h | ⊢ (𝜑 → ran (Homf ‘𝐶) ⊆ 𝑈) |
Ref | Expression |
---|---|
oppchofcl | ⊢ (𝜑 → 𝑀 ∈ ((𝐶 ×c 𝑂) Func 𝐷)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oppchofcl.m | . . 3 ⊢ 𝑀 = (HomF‘𝑂) | |
2 | eqid 2725 | . . 3 ⊢ (oppCat‘𝑂) = (oppCat‘𝑂) | |
3 | oppchofcl.d | . . 3 ⊢ 𝐷 = (SetCat‘𝑈) | |
4 | oppchofcl.c | . . . 4 ⊢ (𝜑 → 𝐶 ∈ Cat) | |
5 | oppchofcl.o | . . . . 5 ⊢ 𝑂 = (oppCat‘𝐶) | |
6 | 5 | oppccat 17732 | . . . 4 ⊢ (𝐶 ∈ Cat → 𝑂 ∈ Cat) |
7 | 4, 6 | syl 17 | . . 3 ⊢ (𝜑 → 𝑂 ∈ Cat) |
8 | oppchofcl.u | . . 3 ⊢ (𝜑 → 𝑈 ∈ 𝑉) | |
9 | eqid 2725 | . . . . . . 7 ⊢ (Homf ‘𝐶) = (Homf ‘𝐶) | |
10 | 5, 9 | oppchomf 17730 | . . . . . 6 ⊢ tpos (Homf ‘𝐶) = (Homf ‘𝑂) |
11 | 10 | rneqi 5942 | . . . . 5 ⊢ ran tpos (Homf ‘𝐶) = ran (Homf ‘𝑂) |
12 | relxp 5699 | . . . . . . 7 ⊢ Rel ((Base‘𝐶) × (Base‘𝐶)) | |
13 | eqid 2725 | . . . . . . . . . 10 ⊢ (Base‘𝐶) = (Base‘𝐶) | |
14 | 9, 13 | homffn 17701 | . . . . . . . . 9 ⊢ (Homf ‘𝐶) Fn ((Base‘𝐶) × (Base‘𝐶)) |
15 | 14 | fndmi 6663 | . . . . . . . 8 ⊢ dom (Homf ‘𝐶) = ((Base‘𝐶) × (Base‘𝐶)) |
16 | 15 | releqi 5782 | . . . . . . 7 ⊢ (Rel dom (Homf ‘𝐶) ↔ Rel ((Base‘𝐶) × (Base‘𝐶))) |
17 | 12, 16 | mpbir 230 | . . . . . 6 ⊢ Rel dom (Homf ‘𝐶) |
18 | rntpos 8253 | . . . . . 6 ⊢ (Rel dom (Homf ‘𝐶) → ran tpos (Homf ‘𝐶) = ran (Homf ‘𝐶)) | |
19 | 17, 18 | ax-mp 5 | . . . . 5 ⊢ ran tpos (Homf ‘𝐶) = ran (Homf ‘𝐶) |
20 | 11, 19 | eqtr3i 2755 | . . . 4 ⊢ ran (Homf ‘𝑂) = ran (Homf ‘𝐶) |
21 | oppchofcl.h | . . . 4 ⊢ (𝜑 → ran (Homf ‘𝐶) ⊆ 𝑈) | |
22 | 20, 21 | eqsstrid 4027 | . . 3 ⊢ (𝜑 → ran (Homf ‘𝑂) ⊆ 𝑈) |
23 | 1, 2, 3, 7, 8, 22 | hofcl 18279 | . 2 ⊢ (𝜑 → 𝑀 ∈ (((oppCat‘𝑂) ×c 𝑂) Func 𝐷)) |
24 | 5 | 2oppchomf 17734 | . . . . 5 ⊢ (Homf ‘𝐶) = (Homf ‘(oppCat‘𝑂)) |
25 | 24 | a1i 11 | . . . 4 ⊢ (𝜑 → (Homf ‘𝐶) = (Homf ‘(oppCat‘𝑂))) |
26 | 5 | 2oppccomf 17735 | . . . . 5 ⊢ (compf‘𝐶) = (compf‘(oppCat‘𝑂)) |
27 | 26 | a1i 11 | . . . 4 ⊢ (𝜑 → (compf‘𝐶) = (compf‘(oppCat‘𝑂))) |
28 | eqidd 2726 | . . . 4 ⊢ (𝜑 → (Homf ‘𝑂) = (Homf ‘𝑂)) | |
29 | eqidd 2726 | . . . 4 ⊢ (𝜑 → (compf‘𝑂) = (compf‘𝑂)) | |
30 | 2 | oppccat 17732 | . . . . 5 ⊢ (𝑂 ∈ Cat → (oppCat‘𝑂) ∈ Cat) |
31 | 7, 30 | syl 17 | . . . 4 ⊢ (𝜑 → (oppCat‘𝑂) ∈ Cat) |
32 | 25, 27, 28, 29, 4, 31, 7, 7 | xpcpropd 18228 | . . 3 ⊢ (𝜑 → (𝐶 ×c 𝑂) = ((oppCat‘𝑂) ×c 𝑂)) |
33 | 32 | oveq1d 7438 | . 2 ⊢ (𝜑 → ((𝐶 ×c 𝑂) Func 𝐷) = (((oppCat‘𝑂) ×c 𝑂) Func 𝐷)) |
34 | 23, 33 | eleqtrrd 2828 | 1 ⊢ (𝜑 → 𝑀 ∈ ((𝐶 ×c 𝑂) Func 𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 ⊆ wss 3946 × cxp 5679 dom cdm 5681 ran crn 5682 Rel wrel 5686 ‘cfv 6553 (class class class)co 7423 tpos ctpos 8239 Basecbs 17208 Catccat 17672 Homf chomf 17674 compfccomf 17675 oppCatcoppc 17719 Func cfunc 17868 SetCatcsetc 18092 ×c cxpc 18187 HomFchof 18268 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5368 ax-pr 5432 ax-un 7745 ax-cnex 11210 ax-resscn 11211 ax-1cn 11212 ax-icn 11213 ax-addcl 11214 ax-addrcl 11215 ax-mulcl 11216 ax-mulrcl 11217 ax-mulcom 11218 ax-addass 11219 ax-mulass 11220 ax-distr 11221 ax-i2m1 11222 ax-1ne0 11223 ax-1rid 11224 ax-rnegex 11225 ax-rrecex 11226 ax-cnre 11227 ax-pre-lttri 11228 ax-pre-lttrn 11229 ax-pre-ltadd 11230 ax-pre-mulgt0 11231 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3966 df-nul 4325 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-tp 4637 df-op 4639 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5579 df-eprel 5585 df-po 5593 df-so 5594 df-fr 5636 df-we 5638 df-xp 5687 df-rel 5688 df-cnv 5689 df-co 5690 df-dm 5691 df-rn 5692 df-res 5693 df-ima 5694 df-pred 6311 df-ord 6378 df-on 6379 df-lim 6380 df-suc 6381 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-riota 7379 df-ov 7426 df-oprab 7427 df-mpo 7428 df-om 7876 df-1st 8002 df-2nd 8003 df-tpos 8240 df-frecs 8295 df-wrecs 8326 df-recs 8400 df-rdg 8439 df-1o 8495 df-er 8733 df-map 8856 df-ixp 8926 df-en 8974 df-dom 8975 df-sdom 8976 df-fin 8977 df-pnf 11296 df-mnf 11297 df-xr 11298 df-ltxr 11299 df-le 11300 df-sub 11492 df-neg 11493 df-nn 12260 df-2 12322 df-3 12323 df-4 12324 df-5 12325 df-6 12326 df-7 12327 df-8 12328 df-9 12329 df-n0 12520 df-z 12606 df-dec 12725 df-uz 12870 df-fz 13534 df-struct 17144 df-sets 17161 df-slot 17179 df-ndx 17191 df-base 17209 df-hom 17285 df-cco 17286 df-cat 17676 df-cid 17677 df-homf 17678 df-comf 17679 df-oppc 17720 df-func 17872 df-setc 18093 df-xpc 18191 df-hof 18270 |
This theorem is referenced by: yoncl 18282 yon11 18284 yon12 18285 yon2 18286 yonpropd 18288 |
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