Theorem oduval 18512

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 23	. . . . 5 ⊢ (𝑎 = 𝑂 → 𝑎 = 𝑂)
2		fveq2 6916	. . . . . . 7 ⊢ (𝑎 = 𝑂 → (le‘𝑎) = (le‘𝑂))
3	2	cnveqd 5865	. . . . . 6 ⊢ (𝑎 = 𝑂 → ◡(le‘𝑎) = ◡(le‘𝑂))
4	3	opeq2d 4852	. . . . 5 ⊢ (𝑎 = 𝑂 → ⟨(le‘ndx), ◡(le‘𝑎)⟩ = ⟨(le‘ndx), ◡(le‘𝑂)⟩)
5	1, 4	oveq12d 7469	. . . 4 ⊢ (𝑎 = 𝑂 → (𝑎 sSet ⟨(le‘ndx), ◡(le‘𝑎)⟩) = (𝑂 sSet ⟨(le‘ndx), ◡(le‘𝑂)⟩))
6		df-odu 18511	. . . 4 ⊢ ODual = (𝑎 ∈ V ↦ (𝑎 sSet ⟨(le‘ndx), ◡(le‘𝑎)⟩))
7		ovex 7484	. . . 4 ⊢ (𝑂 sSet ⟨(le‘ndx), ◡(le‘𝑂)⟩) ∈ V
8	5, 6, 7	fvmpt 7024	. . 3 ⊢ (𝑂 ∈ V → (ODual‘𝑂) = (𝑂 sSet ⟨(le‘ndx), ◡(le‘𝑂)⟩))
9		fvprc 6908	. . . 4 ⊢ (¬ 𝑂 ∈ V → (ODual‘𝑂) = ∅)
10		reldmsets 17387	. . . . 5 ⊢ Rel dom sSet
11	10	ovprc1 7490	. . . 4 ⊢ (¬ 𝑂 ∈ V → (𝑂 sSet ⟨(le‘ndx), ◡(le‘𝑂)⟩) = ∅)
12	9, 11	eqtr4d 2801	. . 3 ⊢ (¬ 𝑂 ∈ V → (ODual‘𝑂) = (𝑂 sSet ⟨(le‘ndx), ◡(le‘𝑂)⟩))
13	8, 12	pm2.61i 184	. 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 5864	. . . 4 ⊢ ◡ ≤ = ◡(le‘𝑂)
17	16	opeq2i 4849	. . 3 ⊢ ⟨(le‘ndx), ◡ ≤ ⟩ = ⟨(le‘ndx), ◡(le‘𝑂)⟩
18	17	oveq2i 7462	. 2 ⊢ (𝑂 sSet ⟨(le‘ndx), ◡ ≤ ⟩) = (𝑂 sSet ⟨(le‘ndx), ◡(le‘𝑂)⟩)
19	13, 14, 18	3eqtr4i 2796	1 ⊢ 𝐷 = (𝑂 sSet ⟨(le‘ndx), ◡ ≤ ⟩)

This proof depends on syntax axioms:  ¬ wn 3   = wceq 1570   ∈ wcel 2147  Vcvv 3454  ∅c0 4281  ⟨cop 4840  ^◡ccnv 5657  ‘cfv 6566  (class class class)co 7451   sSet csts 17385  ndxcnx 17415  lecple 17479  ODualcodu 18510

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5255  ax-nul 5266  ax-pr 5398

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-br 5109  df-opab 5174  df-mpt 5193  df-id 5553  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-iota 6520  df-fun 6568  df-fv 6574  df-ov 7454  df-oprab 7455  df-mpo 7456  df-sets 17386  df-odu 18511

This theorem is used by:  oduleval  18513  odubas  18515