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Mirrors > Home > MPE Home > Th. List > 2oppchomf | Structured version Visualization version GIF version |
Description: The double opposite category has the same morphisms as the original category. Intended for use with property lemmas such as monpropd 16397. (Contributed by Mario Carneiro, 3-Jan-2017.) |
Ref | Expression |
---|---|
oppcbas.1 | ⊢ 𝑂 = (oppCat‘𝐶) |
Ref | Expression |
---|---|
2oppchomf | ⊢ (Homf ‘𝐶) = (Homf ‘(oppCat‘𝑂)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2622 | . . . . 5 ⊢ (Homf ‘𝐶) = (Homf ‘𝐶) | |
2 | eqid 2622 | . . . . 5 ⊢ (Base‘𝐶) = (Base‘𝐶) | |
3 | 1, 2 | homffn 16353 | . . . 4 ⊢ (Homf ‘𝐶) Fn ((Base‘𝐶) × (Base‘𝐶)) |
4 | fnrel 5989 | . . . 4 ⊢ ((Homf ‘𝐶) Fn ((Base‘𝐶) × (Base‘𝐶)) → Rel (Homf ‘𝐶)) | |
5 | 3, 4 | ax-mp 5 | . . 3 ⊢ Rel (Homf ‘𝐶) |
6 | relxp 5227 | . . . 4 ⊢ Rel ((Base‘𝐶) × (Base‘𝐶)) | |
7 | fndm 5990 | . . . . . 6 ⊢ ((Homf ‘𝐶) Fn ((Base‘𝐶) × (Base‘𝐶)) → dom (Homf ‘𝐶) = ((Base‘𝐶) × (Base‘𝐶))) | |
8 | 3, 7 | ax-mp 5 | . . . . 5 ⊢ dom (Homf ‘𝐶) = ((Base‘𝐶) × (Base‘𝐶)) |
9 | 8 | releqi 5202 | . . . 4 ⊢ (Rel dom (Homf ‘𝐶) ↔ Rel ((Base‘𝐶) × (Base‘𝐶))) |
10 | 6, 9 | mpbir 221 | . . 3 ⊢ Rel dom (Homf ‘𝐶) |
11 | tpostpos2 7373 | . . 3 ⊢ ((Rel (Homf ‘𝐶) ∧ Rel dom (Homf ‘𝐶)) → tpos tpos (Homf ‘𝐶) = (Homf ‘𝐶)) | |
12 | 5, 10, 11 | mp2an 708 | . 2 ⊢ tpos tpos (Homf ‘𝐶) = (Homf ‘𝐶) |
13 | eqid 2622 | . . 3 ⊢ (oppCat‘𝑂) = (oppCat‘𝑂) | |
14 | oppcbas.1 | . . . 4 ⊢ 𝑂 = (oppCat‘𝐶) | |
15 | 14, 1 | oppchomf 16380 | . . 3 ⊢ tpos (Homf ‘𝐶) = (Homf ‘𝑂) |
16 | 13, 15 | oppchomf 16380 | . 2 ⊢ tpos tpos (Homf ‘𝐶) = (Homf ‘(oppCat‘𝑂)) |
17 | 12, 16 | eqtr3i 2646 | 1 ⊢ (Homf ‘𝐶) = (Homf ‘(oppCat‘𝑂)) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1483 × cxp 5112 dom cdm 5114 Rel wrel 5119 Fn wfn 5883 ‘cfv 5888 tpos ctpos 7351 Basecbs 15857 Homf chomf 16327 oppCatcoppc 16371 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-tpos 7352 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-7 11084 df-8 11085 df-9 11086 df-n0 11293 df-z 11378 df-dec 11494 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-hom 15966 df-cco 15967 df-homf 16331 df-oppc 16372 |
This theorem is referenced by: 2oppccomf 16385 oppcepi 16399 oppchofcl 16900 oppcyon 16909 oyoncl 16910 |
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