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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 6907 | . . . . . . 7 ⊢ (𝑎 = 𝑂 → (le‘𝑎) = (le‘𝑂)) | |
3 | 2 | cnveqd 5889 | . . . . . 6 ⊢ (𝑎 = 𝑂 → ◡(le‘𝑎) = ◡(le‘𝑂)) |
4 | 3 | opeq2d 4885 | . . . . 5 ⊢ (𝑎 = 𝑂 → 〈(le‘ndx), ◡(le‘𝑎)〉 = 〈(le‘ndx), ◡(le‘𝑂)〉) |
5 | 1, 4 | oveq12d 7449 | . . . 4 ⊢ (𝑎 = 𝑂 → (𝑎 sSet 〈(le‘ndx), ◡(le‘𝑎)〉) = (𝑂 sSet 〈(le‘ndx), ◡(le‘𝑂)〉)) |
6 | df-odu 18344 | . . . 4 ⊢ ODual = (𝑎 ∈ V ↦ (𝑎 sSet 〈(le‘ndx), ◡(le‘𝑎)〉)) | |
7 | ovex 7464 | . . . 4 ⊢ (𝑂 sSet 〈(le‘ndx), ◡(le‘𝑂)〉) ∈ V | |
8 | 5, 6, 7 | fvmpt 7016 | . . 3 ⊢ (𝑂 ∈ V → (ODual‘𝑂) = (𝑂 sSet 〈(le‘ndx), ◡(le‘𝑂)〉)) |
9 | fvprc 6899 | . . . 4 ⊢ (¬ 𝑂 ∈ V → (ODual‘𝑂) = ∅) | |
10 | reldmsets 17199 | . . . . 5 ⊢ Rel dom sSet | |
11 | 10 | ovprc1 7470 | . . . 4 ⊢ (¬ 𝑂 ∈ V → (𝑂 sSet 〈(le‘ndx), ◡(le‘𝑂)〉) = ∅) |
12 | 9, 11 | eqtr4d 2778 | . . 3 ⊢ (¬ 𝑂 ∈ V → (ODual‘𝑂) = (𝑂 sSet 〈(le‘ndx), ◡(le‘𝑂)〉)) |
13 | 8, 12 | pm2.61i 182 | . 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 5888 | . . . 4 ⊢ ◡ ≤ = ◡(le‘𝑂) |
17 | 16 | opeq2i 4882 | . . 3 ⊢ 〈(le‘ndx), ◡ ≤ 〉 = 〈(le‘ndx), ◡(le‘𝑂)〉 |
18 | 17 | oveq2i 7442 | . 2 ⊢ (𝑂 sSet 〈(le‘ndx), ◡ ≤ 〉) = (𝑂 sSet 〈(le‘ndx), ◡(le‘𝑂)〉) |
19 | 13, 14, 18 | 3eqtr4i 2773 | 1 ⊢ 𝐷 = (𝑂 sSet 〈(le‘ndx), ◡ ≤ 〉) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1537 ∈ wcel 2106 Vcvv 3478 ∅c0 4339 〈cop 4637 ◡ccnv 5688 ‘cfv 6563 (class class class)co 7431 sSet csts 17197 ndxcnx 17227 lecple 17305 ODualcodu 18343 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-iota 6516 df-fun 6565 df-fv 6571 df-ov 7434 df-oprab 7435 df-mpo 7436 df-sets 17198 df-odu 18344 |
This theorem is referenced by: oduleval 18346 odubas 18348 odubasOLD 18349 |
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