![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 2735 | . . 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 17769 | . . . 4 ⊢ (𝐶 ∈ Cat → 𝑂 ∈ Cat) |
7 | 4, 6 | syl 17 | . . 3 ⊢ (𝜑 → 𝑂 ∈ Cat) |
8 | oppchofcl.u | . . 3 ⊢ (𝜑 → 𝑈 ∈ 𝑉) | |
9 | eqid 2735 | . . . . . . 7 ⊢ (Homf ‘𝐶) = (Homf ‘𝐶) | |
10 | 5, 9 | oppchomf 17767 | . . . . . 6 ⊢ tpos (Homf ‘𝐶) = (Homf ‘𝑂) |
11 | 10 | rneqi 5951 | . . . . 5 ⊢ ran tpos (Homf ‘𝐶) = ran (Homf ‘𝑂) |
12 | relxp 5707 | . . . . . . 7 ⊢ Rel ((Base‘𝐶) × (Base‘𝐶)) | |
13 | eqid 2735 | . . . . . . . . . 10 ⊢ (Base‘𝐶) = (Base‘𝐶) | |
14 | 9, 13 | homffn 17738 | . . . . . . . . 9 ⊢ (Homf ‘𝐶) Fn ((Base‘𝐶) × (Base‘𝐶)) |
15 | 14 | fndmi 6673 | . . . . . . . 8 ⊢ dom (Homf ‘𝐶) = ((Base‘𝐶) × (Base‘𝐶)) |
16 | 15 | releqi 5790 | . . . . . . 7 ⊢ (Rel dom (Homf ‘𝐶) ↔ Rel ((Base‘𝐶) × (Base‘𝐶))) |
17 | 12, 16 | mpbir 231 | . . . . . 6 ⊢ Rel dom (Homf ‘𝐶) |
18 | rntpos 8263 | . . . . . 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 2765 | . . . 4 ⊢ ran (Homf ‘𝑂) = ran (Homf ‘𝐶) |
21 | oppchofcl.h | . . . 4 ⊢ (𝜑 → ran (Homf ‘𝐶) ⊆ 𝑈) | |
22 | 20, 21 | eqsstrid 4044 | . . 3 ⊢ (𝜑 → ran (Homf ‘𝑂) ⊆ 𝑈) |
23 | 1, 2, 3, 7, 8, 22 | hofcl 18316 | . 2 ⊢ (𝜑 → 𝑀 ∈ (((oppCat‘𝑂) ×c 𝑂) Func 𝐷)) |
24 | 5 | 2oppchomf 17771 | . . . . 5 ⊢ (Homf ‘𝐶) = (Homf ‘(oppCat‘𝑂)) |
25 | 24 | a1i 11 | . . . 4 ⊢ (𝜑 → (Homf ‘𝐶) = (Homf ‘(oppCat‘𝑂))) |
26 | 5 | 2oppccomf 17772 | . . . . 5 ⊢ (compf‘𝐶) = (compf‘(oppCat‘𝑂)) |
27 | 26 | a1i 11 | . . . 4 ⊢ (𝜑 → (compf‘𝐶) = (compf‘(oppCat‘𝑂))) |
28 | eqidd 2736 | . . . 4 ⊢ (𝜑 → (Homf ‘𝑂) = (Homf ‘𝑂)) | |
29 | eqidd 2736 | . . . 4 ⊢ (𝜑 → (compf‘𝑂) = (compf‘𝑂)) | |
30 | 2 | oppccat 17769 | . . . . 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 18265 | . . 3 ⊢ (𝜑 → (𝐶 ×c 𝑂) = ((oppCat‘𝑂) ×c 𝑂)) |
33 | 32 | oveq1d 7446 | . 2 ⊢ (𝜑 → ((𝐶 ×c 𝑂) Func 𝐷) = (((oppCat‘𝑂) ×c 𝑂) Func 𝐷)) |
34 | 23, 33 | eleqtrrd 2842 | 1 ⊢ (𝜑 → 𝑀 ∈ ((𝐶 ×c 𝑂) Func 𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2106 ⊆ wss 3963 × cxp 5687 dom cdm 5689 ran crn 5690 Rel wrel 5694 ‘cfv 6563 (class class class)co 7431 tpos ctpos 8249 Basecbs 17245 Catccat 17709 Homf chomf 17711 compfccomf 17712 oppCatcoppc 17756 Func cfunc 17905 SetCatcsetc 18129 ×c cxpc 18224 HomFchof 18305 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-tp 4636 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-tpos 8250 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-er 8744 df-map 8867 df-ixp 8937 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-9 12334 df-n0 12525 df-z 12612 df-dec 12732 df-uz 12877 df-fz 13545 df-struct 17181 df-sets 17198 df-slot 17216 df-ndx 17228 df-base 17246 df-hom 17322 df-cco 17323 df-cat 17713 df-cid 17714 df-homf 17715 df-comf 17716 df-oppc 17757 df-func 17909 df-setc 18130 df-xpc 18228 df-hof 18307 |
This theorem is referenced by: yoncl 18319 yon11 18321 yon12 18322 yon2 18323 yonpropd 18325 |
Copyright terms: Public domain | W3C validator |