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Mirrors > Home > MPE Home > Th. List > oppcinv | Structured version Visualization version GIF version |
Description: An inverse in the opposite category. (Contributed by Mario Carneiro, 3-Jan-2017.) |
Ref | Expression |
---|---|
oppcsect.b | β’ π΅ = (BaseβπΆ) |
oppcsect.o | β’ π = (oppCatβπΆ) |
oppcsect.c | β’ (π β πΆ β Cat) |
oppcsect.x | β’ (π β π β π΅) |
oppcsect.y | β’ (π β π β π΅) |
oppcinv.s | β’ πΌ = (InvβπΆ) |
oppcinv.t | β’ π½ = (Invβπ) |
Ref | Expression |
---|---|
oppcinv | β’ (π β (ππ½π) = (ππΌπ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | incom 4201 | . . 3 β’ ((π(Sectβπ)π) β© β‘(π(Sectβπ)π)) = (β‘(π(Sectβπ)π) β© (π(Sectβπ)π)) | |
2 | oppcsect.b | . . . . . . 7 β’ π΅ = (BaseβπΆ) | |
3 | oppcsect.o | . . . . . . 7 β’ π = (oppCatβπΆ) | |
4 | oppcsect.c | . . . . . . 7 β’ (π β πΆ β Cat) | |
5 | oppcsect.y | . . . . . . 7 β’ (π β π β π΅) | |
6 | oppcsect.x | . . . . . . 7 β’ (π β π β π΅) | |
7 | eqid 2731 | . . . . . . 7 β’ (SectβπΆ) = (SectβπΆ) | |
8 | eqid 2731 | . . . . . . 7 β’ (Sectβπ) = (Sectβπ) | |
9 | 2, 3, 4, 5, 6, 7, 8 | oppcsect2 17731 | . . . . . 6 β’ (π β (π(Sectβπ)π) = β‘(π(SectβπΆ)π)) |
10 | 9 | cnveqd 5875 | . . . . 5 β’ (π β β‘(π(Sectβπ)π) = β‘β‘(π(SectβπΆ)π)) |
11 | eqid 2731 | . . . . . . . 8 β’ (Hom βπΆ) = (Hom βπΆ) | |
12 | eqid 2731 | . . . . . . . 8 β’ (compβπΆ) = (compβπΆ) | |
13 | eqid 2731 | . . . . . . . 8 β’ (IdβπΆ) = (IdβπΆ) | |
14 | 2, 11, 12, 13, 7, 4, 5, 6 | sectss 17704 | . . . . . . 7 β’ (π β (π(SectβπΆ)π) β ((π(Hom βπΆ)π) Γ (π(Hom βπΆ)π))) |
15 | relxp 5694 | . . . . . . 7 β’ Rel ((π(Hom βπΆ)π) Γ (π(Hom βπΆ)π)) | |
16 | relss 5781 | . . . . . . 7 β’ ((π(SectβπΆ)π) β ((π(Hom βπΆ)π) Γ (π(Hom βπΆ)π)) β (Rel ((π(Hom βπΆ)π) Γ (π(Hom βπΆ)π)) β Rel (π(SectβπΆ)π))) | |
17 | 14, 15, 16 | mpisyl 21 | . . . . . 6 β’ (π β Rel (π(SectβπΆ)π)) |
18 | dfrel2 6188 | . . . . . 6 β’ (Rel (π(SectβπΆ)π) β β‘β‘(π(SectβπΆ)π) = (π(SectβπΆ)π)) | |
19 | 17, 18 | sylib 217 | . . . . 5 β’ (π β β‘β‘(π(SectβπΆ)π) = (π(SectβπΆ)π)) |
20 | 10, 19 | eqtrd 2771 | . . . 4 β’ (π β β‘(π(Sectβπ)π) = (π(SectβπΆ)π)) |
21 | 2, 3, 4, 6, 5, 7, 8 | oppcsect2 17731 | . . . 4 β’ (π β (π(Sectβπ)π) = β‘(π(SectβπΆ)π)) |
22 | 20, 21 | ineq12d 4213 | . . 3 β’ (π β (β‘(π(Sectβπ)π) β© (π(Sectβπ)π)) = ((π(SectβπΆ)π) β© β‘(π(SectβπΆ)π))) |
23 | 1, 22 | eqtrid 2783 | . 2 β’ (π β ((π(Sectβπ)π) β© β‘(π(Sectβπ)π)) = ((π(SectβπΆ)π) β© β‘(π(SectβπΆ)π))) |
24 | 3, 2 | oppcbas 17668 | . . 3 β’ π΅ = (Baseβπ) |
25 | oppcinv.t | . . 3 β’ π½ = (Invβπ) | |
26 | 3 | oppccat 17673 | . . . 4 β’ (πΆ β Cat β π β Cat) |
27 | 4, 26 | syl 17 | . . 3 β’ (π β π β Cat) |
28 | 24, 25, 27, 6, 5, 8 | invfval 17711 | . 2 β’ (π β (ππ½π) = ((π(Sectβπ)π) β© β‘(π(Sectβπ)π))) |
29 | oppcinv.s | . . 3 β’ πΌ = (InvβπΆ) | |
30 | 2, 29, 4, 5, 6, 7 | invfval 17711 | . 2 β’ (π β (ππΌπ) = ((π(SectβπΆ)π) β© β‘(π(SectβπΆ)π))) |
31 | 23, 28, 30 | 3eqtr4d 2781 | 1 β’ (π β (ππ½π) = (ππΌπ)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1540 β wcel 2105 β© cin 3947 β wss 3948 Γ cxp 5674 β‘ccnv 5675 Rel wrel 5681 βcfv 6543 (class class class)co 7412 Basecbs 17149 Hom chom 17213 compcco 17214 Catccat 17613 Idccid 17614 oppCatcoppc 17660 Sectcsect 17696 Invcinv 17697 |
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 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7728 ax-cnex 11169 ax-resscn 11170 ax-1cn 11171 ax-icn 11172 ax-addcl 11173 ax-addrcl 11174 ax-mulcl 11175 ax-mulrcl 11176 ax-mulcom 11177 ax-addass 11178 ax-mulass 11179 ax-distr 11180 ax-i2m1 11181 ax-1ne0 11182 ax-1rid 11183 ax-rnegex 11184 ax-rrecex 11185 ax-cnre 11186 ax-pre-lttri 11187 ax-pre-lttrn 11188 ax-pre-ltadd 11189 ax-pre-mulgt0 11190 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7859 df-1st 7978 df-2nd 7979 df-tpos 8214 df-frecs 8269 df-wrecs 8300 df-recs 8374 df-rdg 8413 df-er 8706 df-en 8943 df-dom 8944 df-sdom 8945 df-pnf 11255 df-mnf 11256 df-xr 11257 df-ltxr 11258 df-le 11259 df-sub 11451 df-neg 11452 df-nn 12218 df-2 12280 df-3 12281 df-4 12282 df-5 12283 df-6 12284 df-7 12285 df-8 12286 df-9 12287 df-n0 12478 df-z 12564 df-dec 12683 df-sets 17102 df-slot 17120 df-ndx 17132 df-base 17150 df-hom 17226 df-cco 17227 df-cat 17617 df-cid 17618 df-oppc 17661 df-sect 17699 df-inv 17700 |
This theorem is referenced by: oppciso 17733 episect 17737 |
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