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Mirrors > Home > MPE Home > Th. List > fthoppc | Structured version Visualization version GIF version |
Description: The opposite functor of a faithful functor is also faithful. Proposition 3.43(c) in [Adamek] p. 39. (Contributed by Mario Carneiro, 27-Jan-2017.) |
Ref | Expression |
---|---|
fulloppc.o | β’ π = (oppCatβπΆ) |
fulloppc.p | β’ π = (oppCatβπ·) |
fthoppc.f | β’ (π β πΉ(πΆ Faith π·)πΊ) |
Ref | Expression |
---|---|
fthoppc | β’ (π β πΉ(π Faith π)tpos πΊ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fulloppc.o | . . 3 β’ π = (oppCatβπΆ) | |
2 | fulloppc.p | . . 3 β’ π = (oppCatβπ·) | |
3 | fthoppc.f | . . . 4 β’ (π β πΉ(πΆ Faith π·)πΊ) | |
4 | fthfunc 17866 | . . . . 5 β’ (πΆ Faith π·) β (πΆ Func π·) | |
5 | 4 | ssbri 5186 | . . . 4 β’ (πΉ(πΆ Faith π·)πΊ β πΉ(πΆ Func π·)πΊ) |
6 | 3, 5 | syl 17 | . . 3 β’ (π β πΉ(πΆ Func π·)πΊ) |
7 | 1, 2, 6 | funcoppc 17831 | . 2 β’ (π β πΉ(π Func π)tpos πΊ) |
8 | eqid 2726 | . . . . . 6 β’ (BaseβπΆ) = (BaseβπΆ) | |
9 | eqid 2726 | . . . . . 6 β’ (Hom βπΆ) = (Hom βπΆ) | |
10 | eqid 2726 | . . . . . 6 β’ (Hom βπ·) = (Hom βπ·) | |
11 | 3 | adantr 480 | . . . . . 6 β’ ((π β§ (π₯ β (BaseβπΆ) β§ π¦ β (BaseβπΆ))) β πΉ(πΆ Faith π·)πΊ) |
12 | simprr 770 | . . . . . 6 β’ ((π β§ (π₯ β (BaseβπΆ) β§ π¦ β (BaseβπΆ))) β π¦ β (BaseβπΆ)) | |
13 | simprl 768 | . . . . . 6 β’ ((π β§ (π₯ β (BaseβπΆ) β§ π¦ β (BaseβπΆ))) β π₯ β (BaseβπΆ)) | |
14 | 8, 9, 10, 11, 12, 13 | fthf1 17876 | . . . . 5 β’ ((π β§ (π₯ β (BaseβπΆ) β§ π¦ β (BaseβπΆ))) β (π¦πΊπ₯):(π¦(Hom βπΆ)π₯)β1-1β((πΉβπ¦)(Hom βπ·)(πΉβπ₯))) |
15 | df-f1 6541 | . . . . . 6 β’ ((π¦πΊπ₯):(π¦(Hom βπΆ)π₯)β1-1β((πΉβπ¦)(Hom βπ·)(πΉβπ₯)) β ((π¦πΊπ₯):(π¦(Hom βπΆ)π₯)βΆ((πΉβπ¦)(Hom βπ·)(πΉβπ₯)) β§ Fun β‘(π¦πΊπ₯))) | |
16 | 15 | simprbi 496 | . . . . 5 β’ ((π¦πΊπ₯):(π¦(Hom βπΆ)π₯)β1-1β((πΉβπ¦)(Hom βπ·)(πΉβπ₯)) β Fun β‘(π¦πΊπ₯)) |
17 | 14, 16 | syl 17 | . . . 4 β’ ((π β§ (π₯ β (BaseβπΆ) β§ π¦ β (BaseβπΆ))) β Fun β‘(π¦πΊπ₯)) |
18 | ovtpos 8224 | . . . . . 6 β’ (π₯tpos πΊπ¦) = (π¦πΊπ₯) | |
19 | 18 | cnveqi 5867 | . . . . 5 β’ β‘(π₯tpos πΊπ¦) = β‘(π¦πΊπ₯) |
20 | 19 | funeqi 6562 | . . . 4 β’ (Fun β‘(π₯tpos πΊπ¦) β Fun β‘(π¦πΊπ₯)) |
21 | 17, 20 | sylibr 233 | . . 3 β’ ((π β§ (π₯ β (BaseβπΆ) β§ π¦ β (BaseβπΆ))) β Fun β‘(π₯tpos πΊπ¦)) |
22 | 21 | ralrimivva 3194 | . 2 β’ (π β βπ₯ β (BaseβπΆ)βπ¦ β (BaseβπΆ)Fun β‘(π₯tpos πΊπ¦)) |
23 | 1, 8 | oppcbas 17669 | . . 3 β’ (BaseβπΆ) = (Baseβπ) |
24 | 23 | isfth 17873 | . 2 β’ (πΉ(π Faith π)tpos πΊ β (πΉ(π Func π)tpos πΊ β§ βπ₯ β (BaseβπΆ)βπ¦ β (BaseβπΆ)Fun β‘(π₯tpos πΊπ¦))) |
25 | 7, 22, 24 | sylanbrc 582 | 1 β’ (π β πΉ(π Faith π)tpos πΊ) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 395 = wceq 1533 β wcel 2098 βwral 3055 class class class wbr 5141 β‘ccnv 5668 Fun wfun 6530 βΆwf 6532 β1-1βwf1 6533 βcfv 6536 (class class class)co 7404 tpos ctpos 8208 Basecbs 17150 Hom chom 17214 oppCatcoppc 17661 Func cfunc 17810 Faith cfth 17862 |
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 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 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 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6293 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-om 7852 df-1st 7971 df-2nd 7972 df-tpos 8209 df-frecs 8264 df-wrecs 8295 df-recs 8369 df-rdg 8408 df-er 8702 df-map 8821 df-ixp 8891 df-en 8939 df-dom 8940 df-sdom 8941 df-pnf 11251 df-mnf 11252 df-xr 11253 df-ltxr 11254 df-le 11255 df-sub 11447 df-neg 11448 df-nn 12214 df-2 12276 df-3 12277 df-4 12278 df-5 12279 df-6 12280 df-7 12281 df-8 12282 df-9 12283 df-n0 12474 df-z 12560 df-dec 12679 df-sets 17103 df-slot 17121 df-ndx 17133 df-base 17151 df-hom 17227 df-cco 17228 df-cat 17618 df-cid 17619 df-oppc 17662 df-func 17814 df-fth 17864 |
This theorem is referenced by: ffthoppc 17883 fthepi 17887 |
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