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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 17903 | . . . . 5 β’ (πΆ Faith π·) β (πΆ Func π·) | |
5 | 4 | ssbri 5197 | . . . 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 17868 | . 2 β’ (π β πΉ(π Func π)tpos πΊ) |
8 | eqid 2728 | . . . . . 6 β’ (BaseβπΆ) = (BaseβπΆ) | |
9 | eqid 2728 | . . . . . 6 β’ (Hom βπΆ) = (Hom βπΆ) | |
10 | eqid 2728 | . . . . . 6 β’ (Hom βπ·) = (Hom βπ·) | |
11 | 3 | adantr 479 | . . . . . 6 β’ ((π β§ (π₯ β (BaseβπΆ) β§ π¦ β (BaseβπΆ))) β πΉ(πΆ Faith π·)πΊ) |
12 | simprr 771 | . . . . . 6 β’ ((π β§ (π₯ β (BaseβπΆ) β§ π¦ β (BaseβπΆ))) β π¦ β (BaseβπΆ)) | |
13 | simprl 769 | . . . . . 6 β’ ((π β§ (π₯ β (BaseβπΆ) β§ π¦ β (BaseβπΆ))) β π₯ β (BaseβπΆ)) | |
14 | 8, 9, 10, 11, 12, 13 | fthf1 17913 | . . . . 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 6558 | . . . . . 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 495 | . . . . 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 8253 | . . . . . 6 β’ (π₯tpos πΊπ¦) = (π¦πΊπ₯) | |
19 | 18 | cnveqi 5881 | . . . . 5 β’ β‘(π₯tpos πΊπ¦) = β‘(π¦πΊπ₯) |
20 | 19 | funeqi 6579 | . . . 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 3198 | . 2 β’ (π β βπ₯ β (BaseβπΆ)βπ¦ β (BaseβπΆ)Fun β‘(π₯tpos πΊπ¦)) |
23 | 1, 8 | oppcbas 17706 | . . 3 β’ (BaseβπΆ) = (Baseβπ) |
24 | 23 | isfth 17910 | . 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 581 | 1 β’ (π β πΉ(π Faith π)tpos πΊ) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 394 = wceq 1533 β wcel 2098 βwral 3058 class class class wbr 5152 β‘ccnv 5681 Fun wfun 6547 βΆwf 6549 β1-1βwf1 6550 βcfv 6553 (class class class)co 7426 tpos ctpos 8237 Basecbs 17187 Hom chom 17251 oppCatcoppc 17698 Func cfunc 17847 Faith cfth 17899 |
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 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11202 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223 |
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 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 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 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-om 7877 df-1st 7999 df-2nd 8000 df-tpos 8238 df-frecs 8293 df-wrecs 8324 df-recs 8398 df-rdg 8437 df-er 8731 df-map 8853 df-ixp 8923 df-en 8971 df-dom 8972 df-sdom 8973 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-sub 11484 df-neg 11485 df-nn 12251 df-2 12313 df-3 12314 df-4 12315 df-5 12316 df-6 12317 df-7 12318 df-8 12319 df-9 12320 df-n0 12511 df-z 12597 df-dec 12716 df-sets 17140 df-slot 17158 df-ndx 17170 df-base 17188 df-hom 17264 df-cco 17265 df-cat 17655 df-cid 17656 df-oppc 17699 df-func 17851 df-fth 17901 |
This theorem is referenced by: ffthoppc 17920 fthepi 17924 |
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