Theorem opprval 20619

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 6916	. . . . . . . 8 ⊢ (𝑥 = 𝑅 → (.r‘𝑥) = (.r‘𝑅))
4		opprval.2	. . . . . . . 8 ⊢ · = (.r‘𝑅)
5	3, 4	eqtr4di 2816	. . . . . . 7 ⊢ (𝑥 = 𝑅 → (.r‘𝑥) = · )
6	5	tposeqd 8284	. . . . . 6 ⊢ (𝑥 = 𝑅 → tpos (.r‘𝑥) = tpos · )
7	6	opeq2d 4852	. . . . 5 ⊢ (𝑥 = 𝑅 → ⟨(.r‘ndx), tpos (.r‘𝑥)⟩ = ⟨(.r‘ndx), tpos · ⟩)
8	2, 7	oveq12d 7469	. . . 4 ⊢ (𝑥 = 𝑅 → (𝑥 sSet ⟨(.r‘ndx), tpos (.r‘𝑥)⟩) = (𝑅 sSet ⟨(.r‘ndx), tpos · ⟩))
9		df-oppr 20618	. . . 4 ⊢ oppr = (𝑥 ∈ V ↦ (𝑥 sSet ⟨(.r‘ndx), tpos (.r‘𝑥)⟩))
10		ovex 7484	. . . 4 ⊢ (𝑅 sSet ⟨(.r‘ndx), tpos · ⟩) ∈ V
11	8, 9, 10	fvmpt 7024	. . 3 ⊢ (𝑅 ∈ V → (oppr‘𝑅) = (𝑅 sSet ⟨(.r‘ndx), tpos · ⟩))
12		fvprc 6908	. . . 4 ⊢ (¬ 𝑅 ∈ V → (oppr‘𝑅) = ∅)
13		reldmsets 17387	. . . . 5 ⊢ Rel dom sSet
14	13	ovprc1 7490	. . . 4 ⊢ (¬ 𝑅 ∈ V → (𝑅 sSet ⟨(.r‘ndx), tpos · ⟩) = ∅)
15	12, 14	eqtr4d 2801	. . 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 2786	1 ⊢ 𝑂 = (𝑅 sSet ⟨(.r‘ndx), tpos · ⟩)

This proof depends on syntax axioms:  ¬ wn 3   = wceq 1570   ∈ wcel 2147  Vcvv 3454  ∅c0 4281  ⟨cop 4840  ‘cfv 6566  (class class class)co 7451  tpos ctpos 8280   sSet csts 17385  ndxcnx 17415  Basecbs 17431  ._rcmulr 17473  opp_rcoppr 20617

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-res 5670  df-iota 6520  df-fun 6568  df-fv 6574  df-ov 7454  df-oprab 7455  df-mpo 7456  df-tpos 8281  df-sets 17386  df-oppr 20618

This theorem is used by:  opprmulfval  20620  opprlem  20623  opprabs  34110