Theorem oduval 18211

Description: Value of an order dual structure. (Contributed by Stefan O'Rear, 29-Jan-2015.)

Hypotheses

Ref	Expression
oduval.d	⊢ 𝐷 = (ODual‘𝑂)
oduval.l	⊢ ≤ = (le‘𝑂)

Assertion

Ref	Expression
oduval	⊢ 𝐷 = (𝑂 sSet ⟨(le‘ndx), ◡ ≤ ⟩)

Proof of Theorem oduval

Dummy variable 𝑎 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		id 22	. . . . 5 ⊢ (𝑎 = 𝑂 → 𝑎 = 𝑂)
2		fveq2 6834	. . . . . . 7 ⊢ (𝑎 = 𝑂 → (le‘𝑎) = (le‘𝑂))
3	2	cnveqd 5824	. . . . . 6 ⊢ (𝑎 = 𝑂 → ◡(le‘𝑎) = ◡(le‘𝑂))
4	3	opeq2d 4836	. . . . 5 ⊢ (𝑎 = 𝑂 → ⟨(le‘ndx), ◡(le‘𝑎)⟩ = ⟨(le‘ndx), ◡(le‘𝑂)⟩)
5	1, 4	oveq12d 7376	. . . 4 ⊢ (𝑎 = 𝑂 → (𝑎 sSet ⟨(le‘ndx), ◡(le‘𝑎)⟩) = (𝑂 sSet ⟨(le‘ndx), ◡(le‘𝑂)⟩))
6		df-odu 18210	. . . 4 ⊢ ODual = (𝑎 ∈ V ↦ (𝑎 sSet ⟨(le‘ndx), ◡(le‘𝑎)⟩))
7		ovex 7391	. . . 4 ⊢ (𝑂 sSet ⟨(le‘ndx), ◡(le‘𝑂)⟩) ∈ V
8	5, 6, 7	fvmpt 6941	. . 3 ⊢ (𝑂 ∈ V → (ODual‘𝑂) = (𝑂 sSet ⟨(le‘ndx), ◡(le‘𝑂)⟩))
9		fvprc 6826	. . . 4 ⊢ (¬ 𝑂 ∈ V → (ODual‘𝑂) = ∅)
10		reldmsets 17092	. . . . 5 ⊢ Rel dom sSet
11	10	ovprc1 7397	. . . 4 ⊢ (¬ 𝑂 ∈ V → (𝑂 sSet ⟨(le‘ndx), ◡(le‘𝑂)⟩) = ∅)
12	9, 11	eqtr4d 2774	. . 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 5823	. . . 4 ⊢ ◡ ≤ = ◡(le‘𝑂)
17	16	opeq2i 4833	. . 3 ⊢ ⟨(le‘ndx), ◡ ≤ ⟩ = ⟨(le‘ndx), ◡(le‘𝑂)⟩
18	17	oveq2i 7369	. 2 ⊢ (𝑂 sSet ⟨(le‘ndx), ◡ ≤ ⟩) = (𝑂 sSet ⟨(le‘ndx), ◡(le‘𝑂)⟩)
19	13, 14, 18	3eqtr4i 2769	1 ⊢ 𝐷 = (𝑂 sSet ⟨(le‘ndx), ◡ ≤ ⟩)

Syntax hints:  ¬ wn 3   = wceq 1541   ∈ wcel 2113  Vcvv 3440  ∅c0 4285  ⟨cop 4586  ^◡ccnv 5623  ‘cfv 6492  (class class class)co 7358   sSet csts 17090  ndxcnx 17120  lecple 17184  ODualcodu 18209

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pr 5377

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-dif 3904  df-un 3906  df-ss 3918  df-nul 4286  df-if 4480  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-br 5099  df-opab 5161  df-mpt 5180  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-iota 6448  df-fun 6494  df-fv 6500  df-ov 7361  df-oprab 7362  df-mpo 7363  df-sets 17091  df-odu 18210

This theorem is referenced by:  oduleval  18212  odubas  18214