Theorem opprval 20416

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 6881	. . . . . . . 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 8221	. . . . . 6 ⊢ (𝑥 = 𝑅 → tpos (.r‘𝑥) = tpos · )
7	6	opeq2d 4845	. . . . 5 ⊢ (𝑥 = 𝑅 → ⟨(.r‘ndx), tpos (.r‘𝑥)⟩ = ⟨(.r‘ndx), tpos · ⟩)
8	2, 7	oveq12d 7428	. . . 4 ⊢ (𝑥 = 𝑅 → (𝑥 sSet ⟨(.r‘ndx), tpos (.r‘𝑥)⟩) = (𝑅 sSet ⟨(.r‘ndx), tpos · ⟩))
9		df-oppr 20415	. . . 4 ⊢ oppr = (𝑥 ∈ V ↦ (𝑥 sSet ⟨(.r‘ndx), tpos (.r‘𝑥)⟩))
10		ovex 7443	. . . 4 ⊢ (𝑅 sSet ⟨(.r‘ndx), tpos · ⟩) ∈ V
11	8, 9, 10	fvmpt 6989	. . 3 ⊢ (𝑅 ∈ V → (oppr‘𝑅) = (𝑅 sSet ⟨(.r‘ndx), tpos · ⟩))
12		fvprc 6873	. . . 4 ⊢ (¬ 𝑅 ∈ V → (oppr‘𝑅) = ∅)
13		reldmsets 17220	. . . . 5 ⊢ Rel dom sSet
14	13	ovprc1 7449	. . . 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 · ⟩)

Syntax hints:  ¬ wn 3   = wceq 1570   ∈ wcel 2143  Vcvv 3455  ∅c0 4286  ⟨cop 4595  ‘cfv 6536  (class class class)co 7410  tpos ctpos 8217   sSet csts 17218  ndxcnx 17248  Basecbs 17264  ._rcmulr 17306  opp_rcoppr 20414

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5257  ax-nul 5269  ax-pr 5404

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-br 5110  df-opab 5174  df-mpt 5193  df-id 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-res 5673  df-iota 6492  df-fun 6538  df-fv 6544  df-ov 7413  df-oprab 7414  df-mpo 7415  df-tpos 8218  df-sets 17219  df-oppr 20415

This theorem is referenced by:  opprmulfval  20417  opprlem  20420  opprabs  33764