Theorem oduval 18455

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 6883	. . . . . . 7 ⊢ (𝑎 = 𝑂 → (le‘𝑎) = (le‘𝑂))
3	2	cnveqd 5853	. . . . . 6 ⊢ (𝑎 = 𝑂 → ◡(le‘𝑎) = ◡(le‘𝑂))
4	3	opeq2d 4840	. . . . 5 ⊢ (𝑎 = 𝑂 → ⟨(le‘ndx), ◡(le‘𝑎)⟩ = ⟨(le‘ndx), ◡(le‘𝑂)⟩)
5	1, 4	oveq12d 7436	. . . 4 ⊢ (𝑎 = 𝑂 → (𝑎 sSet ⟨(le‘ndx), ◡(le‘𝑎)⟩) = (𝑂 sSet ⟨(le‘ndx), ◡(le‘𝑂)⟩))
6		df-odu 18454	. . . 4 ⊢ ODual = (𝑎 ∈ V ↦ (𝑎 sSet ⟨(le‘ndx), ◡(le‘𝑎)⟩))
7		ovex 7451	. . . 4 ⊢ (𝑂 sSet ⟨(le‘ndx), ◡(le‘𝑂)⟩) ∈ V
8	5, 6, 7	fvmpt 6991	. . 3 ⊢ (𝑂 ∈ V → (ODual‘𝑂) = (𝑂 sSet ⟨(le‘ndx), ◡(le‘𝑂)⟩))
9		fvprc 6875	. . . 4 ⊢ (¬ 𝑂 ∈ V → (ODual‘𝑂) = ∅)
10		reldmsets 17336	. . . . 5 ⊢ Rel dom sSet
11	10	ovprc1 7457	. . . 4 ⊢ (¬ 𝑂 ∈ V → (𝑂 sSet ⟨(le‘ndx), ◡(le‘𝑂)⟩) = ∅)
12	9, 11	eqtr4d 2799	. . 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 5852	. . . 4 ⊢ ◡ ≤ = ◡(le‘𝑂)
17	16	opeq2i 4837	. . 3 ⊢ ⟨(le‘ndx), ◡ ≤ ⟩ = ⟨(le‘ndx), ◡(le‘𝑂)⟩
18	17	oveq2i 7429	. 2 ⊢ (𝑂 sSet ⟨(le‘ndx), ◡ ≤ ⟩) = (𝑂 sSet ⟨(le‘ndx), ◡(le‘𝑂)⟩)
19	13, 14, 18	3eqtr4i 2794	1 ⊢ 𝐷 = (𝑂 sSet ⟨(le‘ndx), ◡ ≤ ⟩)

This proof depends on syntax axioms:  ¬ wn 3   = wceq 1570   ∈ wcel 2145  Vcvv 3451  ∅c0 4279  ⟨cop 4590  ^◡ccnv 5650  ‘cfv 6537  (class class class)co 7418   sSet csts 17334  ndxcnx 17364  lecple 17428  ODualcodu 18453

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 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-sep 5249  ax-nul 5260  ax-pr 5391

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 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-iota 6493  df-fun 6539  df-fv 6545  df-ov 7421  df-oprab 7422  df-mpo 7423  df-sets 17335  df-odu 18454

This theorem is used by:  oduleval  18456  odubas  18458