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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 17799 | . . . . 5 β’ (πΆ Faith π·) β (πΆ Func π·) | |
5 | 4 | ssbri 5151 | . . . 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 17766 | . 2 β’ (π β πΉ(π Func π)tpos πΊ) |
8 | eqid 2733 | . . . . . 6 β’ (BaseβπΆ) = (BaseβπΆ) | |
9 | eqid 2733 | . . . . . 6 β’ (Hom βπΆ) = (Hom βπΆ) | |
10 | eqid 2733 | . . . . . 6 β’ (Hom βπ·) = (Hom βπ·) | |
11 | 3 | adantr 482 | . . . . . 6 β’ ((π β§ (π₯ β (BaseβπΆ) β§ π¦ β (BaseβπΆ))) β πΉ(πΆ Faith π·)πΊ) |
12 | simprr 772 | . . . . . 6 β’ ((π β§ (π₯ β (BaseβπΆ) β§ π¦ β (BaseβπΆ))) β π¦ β (BaseβπΆ)) | |
13 | simprl 770 | . . . . . 6 β’ ((π β§ (π₯ β (BaseβπΆ) β§ π¦ β (BaseβπΆ))) β π₯ β (BaseβπΆ)) | |
14 | 8, 9, 10, 11, 12, 13 | fthf1 17809 | . . . . 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 6502 | . . . . . 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 498 | . . . . 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 8173 | . . . . . 6 β’ (π₯tpos πΊπ¦) = (π¦πΊπ₯) | |
19 | 18 | cnveqi 5831 | . . . . 5 β’ β‘(π₯tpos πΊπ¦) = β‘(π¦πΊπ₯) |
20 | 19 | funeqi 6523 | . . . 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 17604 | . . 3 β’ (BaseβπΆ) = (Baseβπ) |
24 | 23 | isfth 17806 | . 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 584 | 1 β’ (π β πΉ(π Faith π)tpos πΊ) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 397 = wceq 1542 β wcel 2107 βwral 3061 class class class wbr 5106 β‘ccnv 5633 Fun wfun 6491 βΆwf 6493 β1-1βwf1 6494 βcfv 6497 (class class class)co 7358 tpos ctpos 8157 Basecbs 17088 Hom chom 17149 oppCatcoppc 17596 Func cfunc 17745 Faith cfth 17795 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5243 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-cnex 11112 ax-resscn 11113 ax-1cn 11114 ax-icn 11115 ax-addcl 11116 ax-addrcl 11117 ax-mulcl 11118 ax-mulrcl 11119 ax-mulcom 11120 ax-addass 11121 ax-mulass 11122 ax-distr 11123 ax-i2m1 11124 ax-1ne0 11125 ax-1rid 11126 ax-rnegex 11127 ax-rrecex 11128 ax-cnre 11129 ax-pre-lttri 11130 ax-pre-lttrn 11131 ax-pre-ltadd 11132 ax-pre-mulgt0 11133 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7804 df-1st 7922 df-2nd 7923 df-tpos 8158 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-er 8651 df-map 8770 df-ixp 8839 df-en 8887 df-dom 8888 df-sdom 8889 df-pnf 11196 df-mnf 11197 df-xr 11198 df-ltxr 11199 df-le 11200 df-sub 11392 df-neg 11393 df-nn 12159 df-2 12221 df-3 12222 df-4 12223 df-5 12224 df-6 12225 df-7 12226 df-8 12227 df-9 12228 df-n0 12419 df-z 12505 df-dec 12624 df-sets 17041 df-slot 17059 df-ndx 17071 df-base 17089 df-hom 17162 df-cco 17163 df-cat 17553 df-cid 17554 df-oppc 17597 df-func 17749 df-fth 17797 |
This theorem is referenced by: ffthoppc 17816 fthepi 17820 |
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