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Theorem oduprs 32129
Description: Being a proset is a self-dual property. (Contributed by Thierry Arnoux, 13-Sep-2018.)
Hypothesis
Ref Expression
oduprs.d 𝐷 = (ODualβ€˜πΎ)
Assertion
Ref Expression
oduprs (𝐾 ∈ Proset β†’ 𝐷 ∈ Proset )

Proof of Theorem oduprs
Dummy variables π‘₯ 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2732 . . . . . . . . . . . . . 14 (Baseβ€˜πΎ) = (Baseβ€˜πΎ)
2 eqid 2732 . . . . . . . . . . . . . 14 (leβ€˜πΎ) = (leβ€˜πΎ)
31, 2isprs 18249 . . . . . . . . . . . . 13 (𝐾 ∈ Proset ↔ (𝐾 ∈ V ∧ βˆ€π‘₯ ∈ (Baseβ€˜πΎ)βˆ€π‘¦ ∈ (Baseβ€˜πΎ)βˆ€π‘§ ∈ (Baseβ€˜πΎ)(π‘₯(leβ€˜πΎ)π‘₯ ∧ ((π‘₯(leβ€˜πΎ)𝑦 ∧ 𝑦(leβ€˜πΎ)𝑧) β†’ π‘₯(leβ€˜πΎ)𝑧))))
43simprbi 497 . . . . . . . . . . . 12 (𝐾 ∈ Proset β†’ βˆ€π‘₯ ∈ (Baseβ€˜πΎ)βˆ€π‘¦ ∈ (Baseβ€˜πΎ)βˆ€π‘§ ∈ (Baseβ€˜πΎ)(π‘₯(leβ€˜πΎ)π‘₯ ∧ ((π‘₯(leβ€˜πΎ)𝑦 ∧ 𝑦(leβ€˜πΎ)𝑧) β†’ π‘₯(leβ€˜πΎ)𝑧)))
54r19.21bi 3248 . . . . . . . . . . 11 ((𝐾 ∈ Proset ∧ π‘₯ ∈ (Baseβ€˜πΎ)) β†’ βˆ€π‘¦ ∈ (Baseβ€˜πΎ)βˆ€π‘§ ∈ (Baseβ€˜πΎ)(π‘₯(leβ€˜πΎ)π‘₯ ∧ ((π‘₯(leβ€˜πΎ)𝑦 ∧ 𝑦(leβ€˜πΎ)𝑧) β†’ π‘₯(leβ€˜πΎ)𝑧)))
65r19.21bi 3248 . . . . . . . . . 10 (((𝐾 ∈ Proset ∧ π‘₯ ∈ (Baseβ€˜πΎ)) ∧ 𝑦 ∈ (Baseβ€˜πΎ)) β†’ βˆ€π‘§ ∈ (Baseβ€˜πΎ)(π‘₯(leβ€˜πΎ)π‘₯ ∧ ((π‘₯(leβ€˜πΎ)𝑦 ∧ 𝑦(leβ€˜πΎ)𝑧) β†’ π‘₯(leβ€˜πΎ)𝑧)))
76r19.21bi 3248 . . . . . . . . 9 ((((𝐾 ∈ Proset ∧ π‘₯ ∈ (Baseβ€˜πΎ)) ∧ 𝑦 ∈ (Baseβ€˜πΎ)) ∧ 𝑧 ∈ (Baseβ€˜πΎ)) β†’ (π‘₯(leβ€˜πΎ)π‘₯ ∧ ((π‘₯(leβ€˜πΎ)𝑦 ∧ 𝑦(leβ€˜πΎ)𝑧) β†’ π‘₯(leβ€˜πΎ)𝑧)))
87simpld 495 . . . . . . . 8 ((((𝐾 ∈ Proset ∧ π‘₯ ∈ (Baseβ€˜πΎ)) ∧ 𝑦 ∈ (Baseβ€˜πΎ)) ∧ 𝑧 ∈ (Baseβ€˜πΎ)) β†’ π‘₯(leβ€˜πΎ)π‘₯)
9 vex 3478 . . . . . . . . 9 π‘₯ ∈ V
109, 9brcnv 5882 . . . . . . . 8 (π‘₯β—‘(leβ€˜πΎ)π‘₯ ↔ π‘₯(leβ€˜πΎ)π‘₯)
118, 10sylibr 233 . . . . . . 7 ((((𝐾 ∈ Proset ∧ π‘₯ ∈ (Baseβ€˜πΎ)) ∧ 𝑦 ∈ (Baseβ€˜πΎ)) ∧ 𝑧 ∈ (Baseβ€˜πΎ)) β†’ π‘₯β—‘(leβ€˜πΎ)π‘₯)
121, 2isprs 18249 . . . . . . . . . . . . . . . . . 18 (𝐾 ∈ Proset ↔ (𝐾 ∈ V ∧ βˆ€π‘§ ∈ (Baseβ€˜πΎ)βˆ€π‘¦ ∈ (Baseβ€˜πΎ)βˆ€π‘₯ ∈ (Baseβ€˜πΎ)(𝑧(leβ€˜πΎ)𝑧 ∧ ((𝑧(leβ€˜πΎ)𝑦 ∧ 𝑦(leβ€˜πΎ)π‘₯) β†’ 𝑧(leβ€˜πΎ)π‘₯))))
1312simprbi 497 . . . . . . . . . . . . . . . . 17 (𝐾 ∈ Proset β†’ βˆ€π‘§ ∈ (Baseβ€˜πΎ)βˆ€π‘¦ ∈ (Baseβ€˜πΎ)βˆ€π‘₯ ∈ (Baseβ€˜πΎ)(𝑧(leβ€˜πΎ)𝑧 ∧ ((𝑧(leβ€˜πΎ)𝑦 ∧ 𝑦(leβ€˜πΎ)π‘₯) β†’ 𝑧(leβ€˜πΎ)π‘₯)))
1413r19.21bi 3248 . . . . . . . . . . . . . . . 16 ((𝐾 ∈ Proset ∧ 𝑧 ∈ (Baseβ€˜πΎ)) β†’ βˆ€π‘¦ ∈ (Baseβ€˜πΎ)βˆ€π‘₯ ∈ (Baseβ€˜πΎ)(𝑧(leβ€˜πΎ)𝑧 ∧ ((𝑧(leβ€˜πΎ)𝑦 ∧ 𝑦(leβ€˜πΎ)π‘₯) β†’ 𝑧(leβ€˜πΎ)π‘₯)))
1514r19.21bi 3248 . . . . . . . . . . . . . . 15 (((𝐾 ∈ Proset ∧ 𝑧 ∈ (Baseβ€˜πΎ)) ∧ 𝑦 ∈ (Baseβ€˜πΎ)) β†’ βˆ€π‘₯ ∈ (Baseβ€˜πΎ)(𝑧(leβ€˜πΎ)𝑧 ∧ ((𝑧(leβ€˜πΎ)𝑦 ∧ 𝑦(leβ€˜πΎ)π‘₯) β†’ 𝑧(leβ€˜πΎ)π‘₯)))
1615r19.21bi 3248 . . . . . . . . . . . . . 14 ((((𝐾 ∈ Proset ∧ 𝑧 ∈ (Baseβ€˜πΎ)) ∧ 𝑦 ∈ (Baseβ€˜πΎ)) ∧ π‘₯ ∈ (Baseβ€˜πΎ)) β†’ (𝑧(leβ€˜πΎ)𝑧 ∧ ((𝑧(leβ€˜πΎ)𝑦 ∧ 𝑦(leβ€˜πΎ)π‘₯) β†’ 𝑧(leβ€˜πΎ)π‘₯)))
1716simprd 496 . . . . . . . . . . . . 13 ((((𝐾 ∈ Proset ∧ 𝑧 ∈ (Baseβ€˜πΎ)) ∧ 𝑦 ∈ (Baseβ€˜πΎ)) ∧ π‘₯ ∈ (Baseβ€˜πΎ)) β†’ ((𝑧(leβ€˜πΎ)𝑦 ∧ 𝑦(leβ€˜πΎ)π‘₯) β†’ 𝑧(leβ€˜πΎ)π‘₯))
1817an32s 650 . . . . . . . . . . . 12 ((((𝐾 ∈ Proset ∧ 𝑧 ∈ (Baseβ€˜πΎ)) ∧ π‘₯ ∈ (Baseβ€˜πΎ)) ∧ 𝑦 ∈ (Baseβ€˜πΎ)) β†’ ((𝑧(leβ€˜πΎ)𝑦 ∧ 𝑦(leβ€˜πΎ)π‘₯) β†’ 𝑧(leβ€˜πΎ)π‘₯))
1918ex 413 . . . . . . . . . . 11 (((𝐾 ∈ Proset ∧ 𝑧 ∈ (Baseβ€˜πΎ)) ∧ π‘₯ ∈ (Baseβ€˜πΎ)) β†’ (𝑦 ∈ (Baseβ€˜πΎ) β†’ ((𝑧(leβ€˜πΎ)𝑦 ∧ 𝑦(leβ€˜πΎ)π‘₯) β†’ 𝑧(leβ€˜πΎ)π‘₯)))
2019an32s 650 . . . . . . . . . 10 (((𝐾 ∈ Proset ∧ π‘₯ ∈ (Baseβ€˜πΎ)) ∧ 𝑧 ∈ (Baseβ€˜πΎ)) β†’ (𝑦 ∈ (Baseβ€˜πΎ) β†’ ((𝑧(leβ€˜πΎ)𝑦 ∧ 𝑦(leβ€˜πΎ)π‘₯) β†’ 𝑧(leβ€˜πΎ)π‘₯)))
2120imp 407 . . . . . . . . 9 ((((𝐾 ∈ Proset ∧ π‘₯ ∈ (Baseβ€˜πΎ)) ∧ 𝑧 ∈ (Baseβ€˜πΎ)) ∧ 𝑦 ∈ (Baseβ€˜πΎ)) β†’ ((𝑧(leβ€˜πΎ)𝑦 ∧ 𝑦(leβ€˜πΎ)π‘₯) β†’ 𝑧(leβ€˜πΎ)π‘₯))
2221an32s 650 . . . . . . . 8 ((((𝐾 ∈ Proset ∧ π‘₯ ∈ (Baseβ€˜πΎ)) ∧ 𝑦 ∈ (Baseβ€˜πΎ)) ∧ 𝑧 ∈ (Baseβ€˜πΎ)) β†’ ((𝑧(leβ€˜πΎ)𝑦 ∧ 𝑦(leβ€˜πΎ)π‘₯) β†’ 𝑧(leβ€˜πΎ)π‘₯))
23 vex 3478 . . . . . . . . . 10 𝑦 ∈ V
249, 23brcnv 5882 . . . . . . . . 9 (π‘₯β—‘(leβ€˜πΎ)𝑦 ↔ 𝑦(leβ€˜πΎ)π‘₯)
25 vex 3478 . . . . . . . . . 10 𝑧 ∈ V
2623, 25brcnv 5882 . . . . . . . . 9 (𝑦◑(leβ€˜πΎ)𝑧 ↔ 𝑧(leβ€˜πΎ)𝑦)
2724, 26anbi12ci 628 . . . . . . . 8 ((π‘₯β—‘(leβ€˜πΎ)𝑦 ∧ 𝑦◑(leβ€˜πΎ)𝑧) ↔ (𝑧(leβ€˜πΎ)𝑦 ∧ 𝑦(leβ€˜πΎ)π‘₯))
289, 25brcnv 5882 . . . . . . . 8 (π‘₯β—‘(leβ€˜πΎ)𝑧 ↔ 𝑧(leβ€˜πΎ)π‘₯)
2922, 27, 283imtr4g 295 . . . . . . 7 ((((𝐾 ∈ Proset ∧ π‘₯ ∈ (Baseβ€˜πΎ)) ∧ 𝑦 ∈ (Baseβ€˜πΎ)) ∧ 𝑧 ∈ (Baseβ€˜πΎ)) β†’ ((π‘₯β—‘(leβ€˜πΎ)𝑦 ∧ 𝑦◑(leβ€˜πΎ)𝑧) β†’ π‘₯β—‘(leβ€˜πΎ)𝑧))
3011, 29jca 512 . . . . . 6 ((((𝐾 ∈ Proset ∧ π‘₯ ∈ (Baseβ€˜πΎ)) ∧ 𝑦 ∈ (Baseβ€˜πΎ)) ∧ 𝑧 ∈ (Baseβ€˜πΎ)) β†’ (π‘₯β—‘(leβ€˜πΎ)π‘₯ ∧ ((π‘₯β—‘(leβ€˜πΎ)𝑦 ∧ 𝑦◑(leβ€˜πΎ)𝑧) β†’ π‘₯β—‘(leβ€˜πΎ)𝑧)))
3130ralrimiva 3146 . . . . 5 (((𝐾 ∈ Proset ∧ π‘₯ ∈ (Baseβ€˜πΎ)) ∧ 𝑦 ∈ (Baseβ€˜πΎ)) β†’ βˆ€π‘§ ∈ (Baseβ€˜πΎ)(π‘₯β—‘(leβ€˜πΎ)π‘₯ ∧ ((π‘₯β—‘(leβ€˜πΎ)𝑦 ∧ 𝑦◑(leβ€˜πΎ)𝑧) β†’ π‘₯β—‘(leβ€˜πΎ)𝑧)))
3231ralrimiva 3146 . . . 4 ((𝐾 ∈ Proset ∧ π‘₯ ∈ (Baseβ€˜πΎ)) β†’ βˆ€π‘¦ ∈ (Baseβ€˜πΎ)βˆ€π‘§ ∈ (Baseβ€˜πΎ)(π‘₯β—‘(leβ€˜πΎ)π‘₯ ∧ ((π‘₯β—‘(leβ€˜πΎ)𝑦 ∧ 𝑦◑(leβ€˜πΎ)𝑧) β†’ π‘₯β—‘(leβ€˜πΎ)𝑧)))
3332ralrimiva 3146 . . 3 (𝐾 ∈ Proset β†’ βˆ€π‘₯ ∈ (Baseβ€˜πΎ)βˆ€π‘¦ ∈ (Baseβ€˜πΎ)βˆ€π‘§ ∈ (Baseβ€˜πΎ)(π‘₯β—‘(leβ€˜πΎ)π‘₯ ∧ ((π‘₯β—‘(leβ€˜πΎ)𝑦 ∧ 𝑦◑(leβ€˜πΎ)𝑧) β†’ π‘₯β—‘(leβ€˜πΎ)𝑧)))
34 oduprs.d . . . 4 𝐷 = (ODualβ€˜πΎ)
3534fvexi 6905 . . 3 𝐷 ∈ V
3633, 35jctil 520 . 2 (𝐾 ∈ Proset β†’ (𝐷 ∈ V ∧ βˆ€π‘₯ ∈ (Baseβ€˜πΎ)βˆ€π‘¦ ∈ (Baseβ€˜πΎ)βˆ€π‘§ ∈ (Baseβ€˜πΎ)(π‘₯β—‘(leβ€˜πΎ)π‘₯ ∧ ((π‘₯β—‘(leβ€˜πΎ)𝑦 ∧ 𝑦◑(leβ€˜πΎ)𝑧) β†’ π‘₯β—‘(leβ€˜πΎ)𝑧))))
3734, 1odubas 18243 . . 3 (Baseβ€˜πΎ) = (Baseβ€˜π·)
3834, 2oduleval 18241 . . 3 β—‘(leβ€˜πΎ) = (leβ€˜π·)
3937, 38isprs 18249 . 2 (𝐷 ∈ Proset ↔ (𝐷 ∈ V ∧ βˆ€π‘₯ ∈ (Baseβ€˜πΎ)βˆ€π‘¦ ∈ (Baseβ€˜πΎ)βˆ€π‘§ ∈ (Baseβ€˜πΎ)(π‘₯β—‘(leβ€˜πΎ)π‘₯ ∧ ((π‘₯β—‘(leβ€˜πΎ)𝑦 ∧ 𝑦◑(leβ€˜πΎ)𝑧) β†’ π‘₯β—‘(leβ€˜πΎ)𝑧))))
4036, 39sylibr 233 1 (𝐾 ∈ Proset β†’ 𝐷 ∈ Proset )
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 396   = wceq 1541   ∈ wcel 2106  βˆ€wral 3061  Vcvv 3474   class class class wbr 5148  β—‘ccnv 5675  β€˜cfv 6543  Basecbs 17143  lecple 17203  ODualcodu 18238   Proset cproset 18245
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7724  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 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  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 7364  df-ov 7411  df-oprab 7412  df-mpo 7413  df-om 7855  df-2nd 7975  df-frecs 8265  df-wrecs 8296  df-recs 8370  df-rdg 8409  df-er 8702  df-en 8939  df-dom 8940  df-sdom 8941  df-pnf 11249  df-mnf 11250  df-xr 11251  df-ltxr 11252  df-le 11253  df-sub 11445  df-neg 11446  df-nn 12212  df-2 12274  df-3 12275  df-4 12276  df-5 12277  df-6 12278  df-7 12279  df-8 12280  df-9 12281  df-dec 12677  df-sets 17096  df-slot 17114  df-ndx 17126  df-base 17144  df-ple 17216  df-odu 18239  df-proset 18247
This theorem is referenced by:  mgccnv  32164  ordtcnvNEW  32895
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