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Mirrors > Home > MPE Home > Th. List > Mathboxes > prstchom2ALT | Structured version Visualization version GIF version |
Description: Hom-sets of the constructed category are dependent on the preorder. This proof depends on the definition df-prstc 47982. See prstchom2 47997 for a version that does not depend on the definition. (Contributed by Zhi Wang, 20-Sep-2024.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
prstcnid.c | β’ (π β πΆ = (ProsetToCatβπΎ)) |
prstcnid.k | β’ (π β πΎ β Proset ) |
prstchom.l | β’ (π β β€ = (leβπΆ)) |
prstchom.e | β’ (π β π» = (Hom βπΆ)) |
Ref | Expression |
---|---|
prstchom2ALT | β’ (π β (π β€ π β β!π π β (ππ»π))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ovex 7447 | . . . 4 β’ (ππ»π) β V | |
2 | prstchom.e | . . . . . . 7 β’ (π β π» = (Hom βπΆ)) | |
3 | prstcnid.c | . . . . . . . 8 β’ (π β πΆ = (ProsetToCatβπΎ)) | |
4 | prstcnid.k | . . . . . . . 8 β’ (π β πΎ β Proset ) | |
5 | prstchom.l | . . . . . . . 8 β’ (π β β€ = (leβπΆ)) | |
6 | 3, 4, 5 | prstchomval 47993 | . . . . . . 7 β’ (π β ( β€ Γ {1o}) = (Hom βπΆ)) |
7 | 2, 6 | eqtr4d 2770 | . . . . . 6 β’ (π β π» = ( β€ Γ {1o})) |
8 | 1oex 8488 | . . . . . . 7 β’ 1o β V | |
9 | 8 | a1i 11 | . . . . . 6 β’ (π β 1o β V) |
10 | 1n0 8500 | . . . . . . 7 β’ 1o β β | |
11 | 10 | a1i 11 | . . . . . 6 β’ (π β 1o β β ) |
12 | 7, 9, 11 | fvconstr 47821 | . . . . 5 β’ (π β (π β€ π β (ππ»π) = 1o)) |
13 | 12 | biimpa 476 | . . . 4 β’ ((π β§ π β€ π) β (ππ»π) = 1o) |
14 | eqeng 8996 | . . . 4 β’ ((ππ»π) β V β ((ππ»π) = 1o β (ππ»π) β 1o)) | |
15 | 1, 13, 14 | mpsyl 68 | . . 3 β’ ((π β§ π β€ π) β (ππ»π) β 1o) |
16 | euen1b 9041 | . . 3 β’ ((ππ»π) β 1o β β!π π β (ππ»π)) | |
17 | 15, 16 | sylib 217 | . 2 β’ ((π β§ π β€ π) β β!π π β (ππ»π)) |
18 | euex 2566 | . . . 4 β’ (β!π π β (ππ»π) β βπ π β (ππ»π)) | |
19 | n0 4342 | . . . 4 β’ ((ππ»π) β β β βπ π β (ππ»π)) | |
20 | 18, 19 | sylibr 233 | . . 3 β’ (β!π π β (ππ»π) β (ππ»π) β β ) |
21 | 7, 9, 11 | fvconstrn0 47822 | . . . 4 β’ (π β (π β€ π β (ππ»π) β β )) |
22 | 21 | biimpar 477 | . . 3 β’ ((π β§ (ππ»π) β β ) β π β€ π) |
23 | 20, 22 | sylan2 592 | . 2 β’ ((π β§ β!π π β (ππ»π)) β π β€ π) |
24 | 17, 23 | impbida 800 | 1 β’ (π β (π β€ π β β!π π β (ππ»π))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 395 = wceq 1534 βwex 1774 β wcel 2099 β!weu 2557 β wne 2935 Vcvv 3469 β c0 4318 {csn 4624 class class class wbr 5142 Γ cxp 5670 βcfv 6542 (class class class)co 7414 1oc1o 8471 β cen 8950 lecple 17225 Hom chom 17229 Proset cproset 18270 ProsetToCatcprstc 47981 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7732 ax-cnex 11180 ax-resscn 11181 ax-1cn 11182 ax-icn 11183 ax-addcl 11184 ax-addrcl 11185 ax-mulcl 11186 ax-mulrcl 11187 ax-mulcom 11188 ax-addass 11189 ax-mulass 11190 ax-distr 11191 ax-i2m1 11192 ax-1ne0 11193 ax-1rid 11194 ax-rnegex 11195 ax-rrecex 11196 ax-cnre 11197 ax-pre-lttri 11198 ax-pre-lttrn 11199 ax-pre-ltadd 11200 ax-pre-mulgt0 11201 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7863 df-2nd 7986 df-frecs 8278 df-wrecs 8309 df-recs 8383 df-rdg 8422 df-1o 8478 df-er 8716 df-en 8954 df-dom 8955 df-sdom 8956 df-pnf 11266 df-mnf 11267 df-xr 11268 df-ltxr 11269 df-le 11270 df-sub 11462 df-neg 11463 df-nn 12229 df-2 12291 df-3 12292 df-4 12293 df-5 12294 df-6 12295 df-7 12296 df-8 12297 df-9 12298 df-n0 12489 df-z 12575 df-dec 12694 df-sets 17118 df-slot 17136 df-ndx 17148 df-base 17166 df-ple 17238 df-hom 17242 df-cco 17243 df-prstc 47982 |
This theorem is referenced by: (None) |
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