Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 6103 | . . . . . . 7 ⊢ (𝑎 = 𝑂 → (le‘𝑎) = (le‘𝑂)) | |
3 | 2 | cnveqd 5220 | . . . . . 6 ⊢ (𝑎 = 𝑂 → ◡(le‘𝑎) = ◡(le‘𝑂)) |
4 | 3 | opeq2d 4347 | . . . . 5 ⊢ (𝑎 = 𝑂 → 〈(le‘ndx), ◡(le‘𝑎)〉 = 〈(le‘ndx), ◡(le‘𝑂)〉) |
5 | 1, 4 | oveq12d 6567 | . . . 4 ⊢ (𝑎 = 𝑂 → (𝑎 sSet 〈(le‘ndx), ◡(le‘𝑎)〉) = (𝑂 sSet 〈(le‘ndx), ◡(le‘𝑂)〉)) |
6 | df-odu 16952 | . . . 4 ⊢ ODual = (𝑎 ∈ V ↦ (𝑎 sSet 〈(le‘ndx), ◡(le‘𝑎)〉)) | |
7 | ovex 6577 | . . . 4 ⊢ (𝑂 sSet 〈(le‘ndx), ◡(le‘𝑂)〉) ∈ V | |
8 | 5, 6, 7 | fvmpt 6191 | . . 3 ⊢ (𝑂 ∈ V → (ODual‘𝑂) = (𝑂 sSet 〈(le‘ndx), ◡(le‘𝑂)〉)) |
9 | fvprc 6097 | . . . 4 ⊢ (¬ 𝑂 ∈ V → (ODual‘𝑂) = ∅) | |
10 | reldmsets 15718 | . . . . 5 ⊢ Rel dom sSet | |
11 | 10 | ovprc1 6582 | . . . 4 ⊢ (¬ 𝑂 ∈ V → (𝑂 sSet 〈(le‘ndx), ◡(le‘𝑂)〉) = ∅) |
12 | 9, 11 | eqtr4d 2647 | . . 3 ⊢ (¬ 𝑂 ∈ V → (ODual‘𝑂) = (𝑂 sSet 〈(le‘ndx), ◡(le‘𝑂)〉)) |
13 | 8, 12 | pm2.61i 175 | . 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 5219 | . . . 4 ⊢ ◡ ≤ = ◡(le‘𝑂) |
17 | 16 | opeq2i 4344 | . . 3 ⊢ 〈(le‘ndx), ◡ ≤ 〉 = 〈(le‘ndx), ◡(le‘𝑂)〉 |
18 | 17 | oveq2i 6560 | . 2 ⊢ (𝑂 sSet 〈(le‘ndx), ◡ ≤ 〉) = (𝑂 sSet 〈(le‘ndx), ◡(le‘𝑂)〉) |
19 | 13, 14, 18 | 3eqtr4i 2642 | 1 ⊢ 𝐷 = (𝑂 sSet 〈(le‘ndx), ◡ ≤ 〉) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1475 ∈ wcel 1977 Vcvv 3173 ∅c0 3874 〈cop 4131 ◡ccnv 5037 ‘cfv 5804 (class class class)co 6549 ndxcnx 15692 sSet csts 15693 lecple 15775 ODualcodu 16951 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-iota 5768 df-fun 5806 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-sets 15701 df-odu 16952 |
This theorem is referenced by: oduleval 16954 odubas 16956 |
Copyright terms: Public domain | W3C validator |