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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 6663 | . . . . . . 7 ⊢ (𝑎 = 𝑂 → (le‘𝑎) = (le‘𝑂)) | |
3 | 2 | cnveqd 5739 | . . . . . 6 ⊢ (𝑎 = 𝑂 → ◡(le‘𝑎) = ◡(le‘𝑂)) |
4 | 3 | opeq2d 4802 | . . . . 5 ⊢ (𝑎 = 𝑂 → 〈(le‘ndx), ◡(le‘𝑎)〉 = 〈(le‘ndx), ◡(le‘𝑂)〉) |
5 | 1, 4 | oveq12d 7163 | . . . 4 ⊢ (𝑎 = 𝑂 → (𝑎 sSet 〈(le‘ndx), ◡(le‘𝑎)〉) = (𝑂 sSet 〈(le‘ndx), ◡(le‘𝑂)〉)) |
6 | df-odu 17727 | . . . 4 ⊢ ODual = (𝑎 ∈ V ↦ (𝑎 sSet 〈(le‘ndx), ◡(le‘𝑎)〉)) | |
7 | ovex 7178 | . . . 4 ⊢ (𝑂 sSet 〈(le‘ndx), ◡(le‘𝑂)〉) ∈ V | |
8 | 5, 6, 7 | fvmpt 6761 | . . 3 ⊢ (𝑂 ∈ V → (ODual‘𝑂) = (𝑂 sSet 〈(le‘ndx), ◡(le‘𝑂)〉)) |
9 | fvprc 6656 | . . . 4 ⊢ (¬ 𝑂 ∈ V → (ODual‘𝑂) = ∅) | |
10 | reldmsets 16499 | . . . . 5 ⊢ Rel dom sSet | |
11 | 10 | ovprc1 7184 | . . . 4 ⊢ (¬ 𝑂 ∈ V → (𝑂 sSet 〈(le‘ndx), ◡(le‘𝑂)〉) = ∅) |
12 | 9, 11 | eqtr4d 2856 | . . 3 ⊢ (¬ 𝑂 ∈ V → (ODual‘𝑂) = (𝑂 sSet 〈(le‘ndx), ◡(le‘𝑂)〉)) |
13 | 8, 12 | pm2.61i 183 | . 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 5738 | . . . 4 ⊢ ◡ ≤ = ◡(le‘𝑂) |
17 | 16 | opeq2i 4799 | . . 3 ⊢ 〈(le‘ndx), ◡ ≤ 〉 = 〈(le‘ndx), ◡(le‘𝑂)〉 |
18 | 17 | oveq2i 7156 | . 2 ⊢ (𝑂 sSet 〈(le‘ndx), ◡ ≤ 〉) = (𝑂 sSet 〈(le‘ndx), ◡(le‘𝑂)〉) |
19 | 13, 14, 18 | 3eqtr4i 2851 | 1 ⊢ 𝐷 = (𝑂 sSet 〈(le‘ndx), ◡ ≤ 〉) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1528 ∈ wcel 2105 Vcvv 3492 ∅c0 4288 〈cop 4563 ◡ccnv 5547 ‘cfv 6348 (class class class)co 7145 ndxcnx 16468 sSet csts 16469 lecple 16560 ODualcodu 17726 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-iota 6307 df-fun 6350 df-fv 6356 df-ov 7148 df-oprab 7149 df-mpo 7150 df-sets 16478 df-odu 17727 |
This theorem is referenced by: oduleval 17729 odubas 17731 |
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