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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 6906 | . . . . . . 7 ⊢ (𝑎 = 𝑂 → (le‘𝑎) = (le‘𝑂)) | |
| 3 | 2 | cnveqd 5886 | . . . . . 6 ⊢ (𝑎 = 𝑂 → ◡(le‘𝑎) = ◡(le‘𝑂)) |
| 4 | 3 | opeq2d 4880 | . . . . 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 18332 | . . . 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 6898 | . . . 4 ⊢ (¬ 𝑂 ∈ V → (ODual‘𝑂) = ∅) | |
| 10 | reldmsets 17202 | . . . . 5 ⊢ Rel dom sSet | |
| 11 | 10 | ovprc1 7470 | . . . 4 ⊢ (¬ 𝑂 ∈ V → (𝑂 sSet 〈(le‘ndx), ◡(le‘𝑂)〉) = ∅) |
| 12 | 9, 11 | eqtr4d 2780 | . . 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 5885 | . . . 4 ⊢ ◡ ≤ = ◡(le‘𝑂) |
| 17 | 16 | opeq2i 4877 | . . 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 2775 | 1 ⊢ 𝐷 = (𝑂 sSet 〈(le‘ndx), ◡ ≤ 〉) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 = wceq 1540 ∈ wcel 2108 Vcvv 3480 ∅c0 4333 〈cop 4632 ◡ccnv 5684 ‘cfv 6561 (class class class)co 7431 sSet csts 17200 ndxcnx 17230 lecple 17304 ODualcodu 18331 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-iota 6514 df-fun 6563 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-sets 17201 df-odu 18332 |
| This theorem is referenced by: oduleval 18334 odubas 18336 odubasOLD 18337 |
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