![]() |
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 6645 | . . . . . . 7 ⊢ (𝑎 = 𝑂 → (le‘𝑎) = (le‘𝑂)) | |
3 | 2 | cnveqd 5710 | . . . . . 6 ⊢ (𝑎 = 𝑂 → ◡(le‘𝑎) = ◡(le‘𝑂)) |
4 | 3 | opeq2d 4772 | . . . . 5 ⊢ (𝑎 = 𝑂 → 〈(le‘ndx), ◡(le‘𝑎)〉 = 〈(le‘ndx), ◡(le‘𝑂)〉) |
5 | 1, 4 | oveq12d 7153 | . . . 4 ⊢ (𝑎 = 𝑂 → (𝑎 sSet 〈(le‘ndx), ◡(le‘𝑎)〉) = (𝑂 sSet 〈(le‘ndx), ◡(le‘𝑂)〉)) |
6 | df-odu 17731 | . . . 4 ⊢ ODual = (𝑎 ∈ V ↦ (𝑎 sSet 〈(le‘ndx), ◡(le‘𝑎)〉)) | |
7 | ovex 7168 | . . . 4 ⊢ (𝑂 sSet 〈(le‘ndx), ◡(le‘𝑂)〉) ∈ V | |
8 | 5, 6, 7 | fvmpt 6745 | . . 3 ⊢ (𝑂 ∈ V → (ODual‘𝑂) = (𝑂 sSet 〈(le‘ndx), ◡(le‘𝑂)〉)) |
9 | fvprc 6638 | . . . 4 ⊢ (¬ 𝑂 ∈ V → (ODual‘𝑂) = ∅) | |
10 | reldmsets 16503 | . . . . 5 ⊢ Rel dom sSet | |
11 | 10 | ovprc1 7174 | . . . 4 ⊢ (¬ 𝑂 ∈ V → (𝑂 sSet 〈(le‘ndx), ◡(le‘𝑂)〉) = ∅) |
12 | 9, 11 | eqtr4d 2836 | . . 3 ⊢ (¬ 𝑂 ∈ V → (ODual‘𝑂) = (𝑂 sSet 〈(le‘ndx), ◡(le‘𝑂)〉)) |
13 | 8, 12 | pm2.61i 185 | . 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 5709 | . . . 4 ⊢ ◡ ≤ = ◡(le‘𝑂) |
17 | 16 | opeq2i 4769 | . . 3 ⊢ 〈(le‘ndx), ◡ ≤ 〉 = 〈(le‘ndx), ◡(le‘𝑂)〉 |
18 | 17 | oveq2i 7146 | . 2 ⊢ (𝑂 sSet 〈(le‘ndx), ◡ ≤ 〉) = (𝑂 sSet 〈(le‘ndx), ◡(le‘𝑂)〉) |
19 | 13, 14, 18 | 3eqtr4i 2831 | 1 ⊢ 𝐷 = (𝑂 sSet 〈(le‘ndx), ◡ ≤ 〉) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1538 ∈ wcel 2111 Vcvv 3441 ∅c0 4243 〈cop 4531 ◡ccnv 5518 ‘cfv 6324 (class class class)co 7135 ndxcnx 16472 sSet csts 16473 lecple 16564 ODualcodu 17730 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-iota 6283 df-fun 6326 df-fv 6332 df-ov 7138 df-oprab 7139 df-mpo 7140 df-sets 16482 df-odu 17731 |
This theorem is referenced by: oduleval 17733 odubas 17735 |
Copyright terms: Public domain | W3C validator |