Theorem opprval 20466

Description: Value of the opposite ring. (Contributed by Mario Carneiro, 1-Dec-2014.)

Hypotheses

Ref	Expression
opprval.1	⊢ 𝐵 = (Base‘𝑅)
opprval.2	⊢ · = (.r‘𝑅)
opprval.3	⊢ 𝑂 = (oppr‘𝑅)

Assertion

Ref	Expression
opprval	⊢ 𝑂 = (𝑅 sSet ⟨(.r‘ndx), tpos · ⟩)

Proof of Theorem opprval

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		opprval.3	. 2 ⊢ 𝑂 = (oppr‘𝑅)
2		id 23	. . . . 5 ⊢ (𝑥 = 𝑅 → 𝑥 = 𝑅)
3		fveq2 6885	. . . . . . . 8 ⊢ (𝑥 = 𝑅 → (.r‘𝑥) = (.r‘𝑅))
4		opprval.2	. . . . . . . 8 ⊢ · = (.r‘𝑅)
5	3, 4	eqtr4di 2818	. . . . . . 7 ⊢ (𝑥 = 𝑅 → (.r‘𝑥) = · )
6	5	tposeqd 8231	. . . . . 6 ⊢ (𝑥 = 𝑅 → tpos (.r‘𝑥) = tpos · )
7	6	opeq2d 4847	. . . . 5 ⊢ (𝑥 = 𝑅 → ⟨(.r‘ndx), tpos (.r‘𝑥)⟩ = ⟨(.r‘ndx), tpos · ⟩)
8	2, 7	oveq12d 7437	. . . 4 ⊢ (𝑥 = 𝑅 → (𝑥 sSet ⟨(.r‘ndx), tpos (.r‘𝑥)⟩) = (𝑅 sSet ⟨(.r‘ndx), tpos · ⟩))
9		df-oppr 20465	. . . 4 ⊢ oppr = (𝑥 ∈ V ↦ (𝑥 sSet ⟨(.r‘ndx), tpos (.r‘𝑥)⟩))
10		ovex 7452	. . . 4 ⊢ (𝑅 sSet ⟨(.r‘ndx), tpos · ⟩) ∈ V
11	8, 9, 10	fvmpt 6993	. . 3 ⊢ (𝑅 ∈ V → (oppr‘𝑅) = (𝑅 sSet ⟨(.r‘ndx), tpos · ⟩))
12		fvprc 6877	. . . 4 ⊢ (¬ 𝑅 ∈ V → (oppr‘𝑅) = ∅)
13		reldmsets 17247	. . . . 5 ⊢ Rel dom sSet
14	13	ovprc1 7458	. . . 4 ⊢ (¬ 𝑅 ∈ V → (𝑅 sSet ⟨(.r‘ndx), tpos · ⟩) = ∅)
15	12, 14	eqtr4d 2803	. . 3 ⊢ (¬ 𝑅 ∈ V → (oppr‘𝑅) = (𝑅 sSet ⟨(.r‘ndx), tpos · ⟩))
16	11, 15	pm2.61i 184	. 2 ⊢ (oppr‘𝑅) = (𝑅 sSet ⟨(.r‘ndx), tpos · ⟩)
17	1, 16	eqtri 2788	1 ⊢ 𝑂 = (𝑅 sSet ⟨(.r‘ndx), tpos · ⟩)

This proof depends on syntax axioms:  ¬ wn 3   = wceq 1570   ∈ wcel 2146  Vcvv 3457  ∅c0 4286  ⟨cop 4597  ‘cfv 6540  (class class class)co 7419  tpos ctpos 8227   sSet csts 17245  ndxcnx 17275  Basecbs 17291  ._rcmulr 17333  opp_rcoppr 20464

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 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-sep 5259  ax-nul 5271  ax-pr 5406

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 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-rab 3419  df-v 3459  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-br 5112  df-opab 5176  df-mpt 5195  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-res 5675  df-iota 6496  df-fun 6542  df-fv 6548  df-ov 7422  df-oprab 7423  df-mpo 7424  df-tpos 8228  df-sets 17246  df-oppr 20465

This theorem is used by:  opprmulfval  20467  opprlem  20470  opprabs  33828