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Mirrors > Home > MPE Home > Th. List > oduval | Structured version Visualization version GIF version |
Description: Value of an order dual structure. (Contributed by Stefan O'Rear, 29-Jan-2015.) |
Ref | Expression |
---|---|
oduval.d | ⊢ 𝐷 = (ODual‘𝑂) |
oduval.l | ⊢ ≤ = (le‘𝑂) |
Ref | Expression |
---|---|
oduval | ⊢ 𝐷 = (𝑂 sSet 〈(le‘ndx), ◡ ≤ 〉) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | id 22 | . . . . 5 ⊢ (𝑎 = 𝑂 → 𝑎 = 𝑂) | |
2 | fveq2 6672 | . . . . . . 7 ⊢ (𝑎 = 𝑂 → (le‘𝑎) = (le‘𝑂)) | |
3 | 2 | cnveqd 5748 | . . . . . 6 ⊢ (𝑎 = 𝑂 → ◡(le‘𝑎) = ◡(le‘𝑂)) |
4 | 3 | opeq2d 4812 | . . . . 5 ⊢ (𝑎 = 𝑂 → 〈(le‘ndx), ◡(le‘𝑎)〉 = 〈(le‘ndx), ◡(le‘𝑂)〉) |
5 | 1, 4 | oveq12d 7176 | . . . 4 ⊢ (𝑎 = 𝑂 → (𝑎 sSet 〈(le‘ndx), ◡(le‘𝑎)〉) = (𝑂 sSet 〈(le‘ndx), ◡(le‘𝑂)〉)) |
6 | df-odu 17741 | . . . 4 ⊢ ODual = (𝑎 ∈ V ↦ (𝑎 sSet 〈(le‘ndx), ◡(le‘𝑎)〉)) | |
7 | ovex 7191 | . . . 4 ⊢ (𝑂 sSet 〈(le‘ndx), ◡(le‘𝑂)〉) ∈ V | |
8 | 5, 6, 7 | fvmpt 6770 | . . 3 ⊢ (𝑂 ∈ V → (ODual‘𝑂) = (𝑂 sSet 〈(le‘ndx), ◡(le‘𝑂)〉)) |
9 | fvprc 6665 | . . . 4 ⊢ (¬ 𝑂 ∈ V → (ODual‘𝑂) = ∅) | |
10 | reldmsets 16513 | . . . . 5 ⊢ Rel dom sSet | |
11 | 10 | ovprc1 7197 | . . . 4 ⊢ (¬ 𝑂 ∈ V → (𝑂 sSet 〈(le‘ndx), ◡(le‘𝑂)〉) = ∅) |
12 | 9, 11 | eqtr4d 2861 | . . 3 ⊢ (¬ 𝑂 ∈ V → (ODual‘𝑂) = (𝑂 sSet 〈(le‘ndx), ◡(le‘𝑂)〉)) |
13 | 8, 12 | pm2.61i 184 | . 2 ⊢ (ODual‘𝑂) = (𝑂 sSet 〈(le‘ndx), ◡(le‘𝑂)〉) |
14 | oduval.d | . 2 ⊢ 𝐷 = (ODual‘𝑂) | |
15 | oduval.l | . . . . 5 ⊢ ≤ = (le‘𝑂) | |
16 | 15 | cnveqi 5747 | . . . 4 ⊢ ◡ ≤ = ◡(le‘𝑂) |
17 | 16 | opeq2i 4809 | . . 3 ⊢ 〈(le‘ndx), ◡ ≤ 〉 = 〈(le‘ndx), ◡(le‘𝑂)〉 |
18 | 17 | oveq2i 7169 | . 2 ⊢ (𝑂 sSet 〈(le‘ndx), ◡ ≤ 〉) = (𝑂 sSet 〈(le‘ndx), ◡(le‘𝑂)〉) |
19 | 13, 14, 18 | 3eqtr4i 2856 | 1 ⊢ 𝐷 = (𝑂 sSet 〈(le‘ndx), ◡ ≤ 〉) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1537 ∈ wcel 2114 Vcvv 3496 ∅c0 4293 〈cop 4575 ◡ccnv 5556 ‘cfv 6357 (class class class)co 7158 ndxcnx 16482 sSet csts 16483 lecple 16574 ODualcodu 17740 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-iota 6316 df-fun 6359 df-fv 6365 df-ov 7161 df-oprab 7162 df-mpo 7163 df-sets 16492 df-odu 17741 |
This theorem is referenced by: oduleval 17743 odubas 17745 |
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