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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 48177. See prstchom2 48192 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 7446 | . . . 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 48188 | . . . . . . 7 β’ (π β ( β€ Γ {1o}) = (Hom βπΆ)) |
7 | 2, 6 | eqtr4d 2768 | . . . . . 6 β’ (π β π» = ( β€ Γ {1o})) |
8 | 1oex 8490 | . . . . . . 7 β’ 1o β V | |
9 | 8 | a1i 11 | . . . . . 6 β’ (π β 1o β V) |
10 | 1n0 8502 | . . . . . . 7 β’ 1o β β | |
11 | 10 | a1i 11 | . . . . . 6 β’ (π β 1o β β ) |
12 | 7, 9, 11 | fvconstr 48016 | . . . . 5 β’ (π β (π β€ π β (ππ»π) = 1o)) |
13 | 12 | biimpa 475 | . . . 4 β’ ((π β§ π β€ π) β (ππ»π) = 1o) |
14 | eqeng 9000 | . . . 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 9045 | . . 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 2565 | . . . 4 β’ (β!π π β (ππ»π) β βπ π β (ππ»π)) | |
19 | n0 4343 | . . . 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 48017 | . . . 4 β’ (π β (π β€ π β (ππ»π) β β )) |
22 | 21 | biimpar 476 | . . 3 β’ ((π β§ (ππ»π) β β ) β π β€ π) |
23 | 20, 22 | sylan2 591 | . 2 β’ ((π β§ β!π π β (ππ»π)) β π β€ π) |
24 | 17, 23 | impbida 799 | 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 394 = wceq 1533 βwex 1773 β wcel 2098 β!weu 2556 β wne 2930 Vcvv 3463 β c0 4319 {csn 4625 class class class wbr 5144 Γ cxp 5671 βcfv 6543 (class class class)co 7413 1oc1o 8473 β cen 8954 lecple 17234 Hom chom 17238 Proset cproset 18279 ProsetToCatcprstc 48176 |
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 2696 ax-sep 5295 ax-nul 5302 ax-pow 5360 ax-pr 5424 ax-un 7735 ax-cnex 11189 ax-resscn 11190 ax-1cn 11191 ax-icn 11192 ax-addcl 11193 ax-addrcl 11194 ax-mulcl 11195 ax-mulrcl 11196 ax-mulcom 11197 ax-addass 11198 ax-mulass 11199 ax-distr 11200 ax-i2m1 11201 ax-1ne0 11202 ax-1rid 11203 ax-rnegex 11204 ax-rrecex 11205 ax-cnre 11206 ax-pre-lttri 11207 ax-pre-lttrn 11208 ax-pre-ltadd 11209 ax-pre-mulgt0 11210 |
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 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3961 df-nul 4320 df-if 4526 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4905 df-iun 4994 df-br 5145 df-opab 5207 df-mpt 5228 df-tr 5262 df-id 5571 df-eprel 5577 df-po 5585 df-so 5586 df-fr 5628 df-we 5630 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-res 5685 df-ima 5686 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 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 7369 df-ov 7416 df-oprab 7417 df-mpo 7418 df-om 7866 df-2nd 7988 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-er 8718 df-en 8958 df-dom 8959 df-sdom 8960 df-pnf 11275 df-mnf 11276 df-xr 11277 df-ltxr 11278 df-le 11279 df-sub 11471 df-neg 11472 df-nn 12238 df-2 12300 df-3 12301 df-4 12302 df-5 12303 df-6 12304 df-7 12305 df-8 12306 df-9 12307 df-n0 12498 df-z 12584 df-dec 12703 df-sets 17127 df-slot 17145 df-ndx 17157 df-base 17175 df-ple 17247 df-hom 17251 df-cco 17252 df-prstc 48177 |
This theorem is referenced by: (None) |
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