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Theorem oduprs 31873
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 2733 . . . . . . . . . . . . . 14 (Baseβ€˜πΎ) = (Baseβ€˜πΎ)
2 eqid 2733 . . . . . . . . . . . . . 14 (leβ€˜πΎ) = (leβ€˜πΎ)
31, 2isprs 18191 . . . . . . . . . . . . 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 498 . . . . . . . . . . . 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 3233 . . . . . . . . . . 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 3233 . . . . . . . . . 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 3233 . . . . . . . . 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 496 . . . . . . . 8 ((((𝐾 ∈ Proset ∧ π‘₯ ∈ (Baseβ€˜πΎ)) ∧ 𝑦 ∈ (Baseβ€˜πΎ)) ∧ 𝑧 ∈ (Baseβ€˜πΎ)) β†’ π‘₯(leβ€˜πΎ)π‘₯)
9 vex 3448 . . . . . . . . 9 π‘₯ ∈ V
109, 9brcnv 5839 . . . . . . . 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 18191 . . . . . . . . . . . . . . . . . 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 498 . . . . . . . . . . . . . . . . 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 3233 . . . . . . . . . . . . . . . 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 3233 . . . . . . . . . . . . . . 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 3233 . . . . . . . . . . . . . 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 497 . . . . . . . . . . . . 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 651 . . . . . . . . . . . 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 414 . . . . . . . . . . 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 651 . . . . . . . . . 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 408 . . . . . . . . 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 651 . . . . . . . 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 3448 . . . . . . . . . 10 𝑦 ∈ V
249, 23brcnv 5839 . . . . . . . . 9 (π‘₯β—‘(leβ€˜πΎ)𝑦 ↔ 𝑦(leβ€˜πΎ)π‘₯)
25 vex 3448 . . . . . . . . . 10 𝑧 ∈ V
2623, 25brcnv 5839 . . . . . . . . 9 (𝑦◑(leβ€˜πΎ)𝑧 ↔ 𝑧(leβ€˜πΎ)𝑦)
2724, 26anbi12ci 629 . . . . . . . 8 ((π‘₯β—‘(leβ€˜πΎ)𝑦 ∧ 𝑦◑(leβ€˜πΎ)𝑧) ↔ (𝑧(leβ€˜πΎ)𝑦 ∧ 𝑦(leβ€˜πΎ)π‘₯))
289, 25brcnv 5839 . . . . . . . 8 (π‘₯β—‘(leβ€˜πΎ)𝑧 ↔ 𝑧(leβ€˜πΎ)π‘₯)
2922, 27, 283imtr4g 296 . . . . . . 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 513 . . . . . 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 3140 . . . . 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 3140 . . . 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 3140 . . 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 6857 . . 3 𝐷 ∈ V
3633, 35jctil 521 . 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 18185 . . 3 (Baseβ€˜πΎ) = (Baseβ€˜π·)
3834, 2oduleval 18183 . . 3 β—‘(leβ€˜πΎ) = (leβ€˜π·)
3937, 38isprs 18191 . 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 397   = wceq 1542   ∈ wcel 2107  βˆ€wral 3061  Vcvv 3444   class class class wbr 5106  β—‘ccnv 5633  β€˜cfv 6497  Basecbs 17088  lecple 17145  ODualcodu 18180   Proset cproset 18187
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-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-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-2nd 7923  df-frecs 8213  df-wrecs 8244  df-recs 8318  df-rdg 8357  df-er 8651  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-dec 12624  df-sets 17041  df-slot 17059  df-ndx 17071  df-base 17089  df-ple 17158  df-odu 18181  df-proset 18189
This theorem is referenced by:  mgccnv  31908  ordtcnvNEW  32558
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