Theorem opbrop 2472

Description: Ordered pair membership in a relation. Special case.

Hypotheses

Ref	Expression
opbrop.1	⊢ (((z = A ∧ w = B) ∧ (v = C ∧ u = D)) → (φ ↔ ψ))
opbrop.2	⊢ R = {⟨x, y⟩∣((x ∈ (S × S) ∧ y ∈ (S × S)) ∧ ∃z∃w∃v∃u((x = ⟨z, w⟩ ∧ y = ⟨v, u⟩) ∧ φ))}

Assertion

Ref	Expression
opbrop	⊢ (((A ∈ S ∧ B ∈ S) ∧ (C ∈ S ∧ D ∈ S)) → (⟨A, B⟩R⟨C, D⟩ ↔ ψ))

Distinct variable group(s):   x,y,z,w,v,u,A   x,B,y,z,w,v,u   x,C,y,z,w,v,u   x,D,y,z,w,v,u   x,S,y,z,w,v,u   φ,x,y   ψ,z,w,v,u

Proof of Theorem opbrop

Step	Hyp	Ref	Expression
1		opbrop.1	. . . . 5 ⊢ (((z = A ∧ w = B) ∧ (v = C ∧ u = D)) → (φ ↔ ψ))
2	1	copsex4g 1904	. . . 4 ⊢ (((A ∈ S ∧ B ∈ S) ∧ (C ∈ S ∧ D ∈ S)) → (∃z∃w∃v∃u((⟨A, B⟩ = ⟨z, w⟩ ∧ ⟨C, D⟩ = ⟨v, u⟩) ∧ φ) ↔ ψ))
3	2	anbi2d 468	. . 3 ⊢ (((A ∈ S ∧ B ∈ S) ∧ (C ∈ S ∧ D ∈ S)) → (((⟨A, B⟩ ∈ (S × S) ∧ ⟨C, D⟩ ∈ (S × S)) ∧ ∃z∃w∃v∃u((⟨A, B⟩ = ⟨z, w⟩ ∧ ⟨C, D⟩ = ⟨v, u⟩) ∧ φ)) ↔ ((⟨A, B⟩ ∈ (S × S) ∧ ⟨C, D⟩ ∈ (S × S)) ∧ ψ)))
4		opex 1893	. . . 4 ⊢ ⟨A, B⟩ ∈ V
5		opex 1893	. . . 4 ⊢ ⟨C, D⟩ ∈ V
6		eleq1 1149	. . . . . 6 ⊢ (x = ⟨A, B⟩ → (x ∈ (S × S) ↔ ⟨A, B⟩ ∈ (S × S)))
7	6	anbi1d 469	. . . . 5 ⊢ (x = ⟨A, B⟩ → ((x ∈ (S × S) ∧ y ∈ (S × S)) ↔ (⟨A, B⟩ ∈ (S × S) ∧ y ∈ (S × S))))
8		cleq1 1107	. . . . . . . 8 ⊢ (x = ⟨A, B⟩ → (x = ⟨z, w⟩ ↔ ⟨A, B⟩ = ⟨z, w⟩))
9	8	anbi1d 469	. . . . . . 7 ⊢ (x = ⟨A, B⟩ → ((x = ⟨z, w⟩ ∧ y = ⟨v, u⟩) ↔ (⟨A, B⟩ = ⟨z, w⟩ ∧ y = ⟨v, u⟩)))
10	9	anbi1d 469	. . . . . 6 ⊢ (x = ⟨A, B⟩ → (((x = ⟨z, w⟩ ∧ y = ⟨v, u⟩) ∧ φ) ↔ ((⟨A, B⟩ = ⟨z, w⟩ ∧ y = ⟨v, u⟩) ∧ φ)))
11	10	bi4exdv 940	. . . . 5 ⊢ (x = ⟨A, B⟩ → (∃z∃w∃v∃u((x = ⟨z, w⟩ ∧ y = ⟨v, u⟩) ∧ φ) ↔ ∃z∃w∃v∃u((⟨A, B⟩ = ⟨z, w⟩ ∧ y = ⟨v, u⟩) ∧ φ)))
12	7, 11	anbi12d 476	. . . 4 ⊢ (x = ⟨A, B⟩ → (((x ∈ (S × S) ∧ y ∈ (S × S)) ∧ ∃z∃w∃v∃u((x = ⟨z, w⟩ ∧ y = ⟨v, u⟩) ∧ φ)) ↔ ((⟨A, B⟩ ∈ (S × S) ∧ y ∈ (S × S)) ∧ ∃z∃w∃v∃u((⟨A, B⟩ = ⟨z, w⟩ ∧ y = ⟨v, u⟩) ∧ φ))))
13		eleq1 1149	. . . . . 6 ⊢ (y = ⟨C, D⟩ → (y ∈ (S × S) ↔ ⟨C, D⟩ ∈ (S × S)))
14	13	anbi2d 468	. . . . 5 ⊢ (y = ⟨C, D⟩ → ((⟨A, B⟩ ∈ (S × S) ∧ y ∈ (S × S)) ↔ (⟨A, B⟩ ∈ (S × S) ∧ ⟨C, D⟩ ∈ (S × S))))
15		cleq1 1107	. . . . . . . 8 ⊢ (y = ⟨C, D⟩ → (y = ⟨v, u⟩ ↔ ⟨C, D⟩ = ⟨v, u⟩))
16	15	anbi2d 468	. . . . . . 7 ⊢ (y = ⟨C, D⟩ → ((⟨A, B⟩ = ⟨z, w⟩ ∧ y = ⟨v, u⟩) ↔ (⟨A, B⟩ = ⟨z, w⟩ ∧ ⟨C, D⟩ = ⟨v, u⟩)))
17	16	anbi1d 469	. . . . . 6 ⊢ (y = ⟨C, D⟩ → (((⟨A, B⟩ = ⟨z, w⟩ ∧ y = ⟨v, u⟩) ∧ φ) ↔ ((⟨A, B⟩ = ⟨z, w⟩ ∧ ⟨C, D⟩ = ⟨v, u⟩) ∧ φ)))
18	17	bi4exdv 940	. . . . 5 ⊢ (y = ⟨C, D⟩ → (∃z∃w∃v∃u((⟨A, B⟩ = ⟨z, w⟩ ∧ y = ⟨v, u⟩) ∧ φ) ↔ ∃z∃w∃v∃u((⟨A, B⟩ = ⟨z, w⟩ ∧ ⟨C, D⟩ = ⟨v, u⟩) ∧ φ)))
19	14, 18	anbi12d 476	. . . 4 ⊢ (y = ⟨C, D⟩ → (((⟨A, B⟩ ∈ (S × S) ∧ y ∈ (S × S)) ∧ ∃z∃w∃v∃u((⟨A, B⟩ = ⟨z, w⟩ ∧ y = ⟨v, u⟩) ∧ φ)) ↔ ((⟨A, B⟩ ∈ (S × S) ∧ ⟨C, D⟩ ∈ (S × S)) ∧ ∃z∃w∃v∃u((⟨A, B⟩ = ⟨z, w⟩ ∧ ⟨C, D⟩ = ⟨v, u⟩) ∧ φ))))
20		opbrop.2	. . . 4 ⊢ R = {⟨x, y⟩∣((x ∈ (S × S) ∧ y ∈ (S × S)) ∧ ∃z∃w∃v∃u((x = ⟨z, w⟩ ∧ y = ⟨v, u⟩) ∧ φ))}
21	4, 5, 12, 19, 20	brab 2118	. . 3 ⊢ (⟨A, B⟩R⟨C, D⟩ ↔ ((⟨A, B⟩ ∈ (S × S) ∧ ⟨C, D⟩ ∈ (S × S)) ∧ ∃z∃w∃v∃u((⟨A, B⟩ = ⟨z, w⟩ ∧ ⟨C, D⟩ = ⟨v, u⟩) ∧ φ)))
22	3, 21	syl5bb 410	. 2 ⊢ (((A ∈ S ∧ B ∈ S) ∧ (C ∈ S ∧ D ∈ S)) → (⟨A, B⟩R⟨C, D⟩ ↔ ((⟨A, B⟩ ∈ (S × S) ∧ ⟨C, D⟩ ∈ (S × S)) ∧ ψ)))
23		opelxpi 2455	. . . 4 ⊢ ((A ∈ S ∧ B ∈ S) → ⟨A, B⟩ ∈ (S × S))
24		opelxpi 2455	. . . 4 ⊢ ((C ∈ S ∧ D ∈ S) → ⟨C, D⟩ ∈ (S × S))
25	23, 24	anim12i 268	. . 3 ⊢ (((A ∈ S ∧ B ∈ S) ∧ (C ∈ S ∧ D ∈ S)) → (⟨A, B⟩ ∈ (S × S) ∧ ⟨C, D⟩ ∈ (S × S)))
26	25	biantrurd 546	. 2 ⊢ (((A ∈ S ∧ B ∈ S) ∧ (C ∈ S ∧ D ∈ S)) → (ψ ↔ ((⟨A, B⟩ ∈ (S × S) ∧ ⟨C, D⟩ ∈ (S × S)) ∧ ψ)))
27	22, 26	bitr4d 409	1 ⊢ (((A ∈ S ∧ B ∈ S) ∧ (C ∈ S ∧ D ∈ S)) → (⟨A, B⟩R⟨C, D⟩ ↔ ψ))

Syntax hints:   → wi 2   ↔ wb 127   ∧ wa 196  ∃wex 678   = wceq 1091   ∈ wcel 1092  ⟨cop 1810   class class class wbr 2054  {copab 2055   × cxp 2408

This theorem is referenced by:  ecopopreq 3244  oprec 3254

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-br 2063  df-opab 2098  df-xp 2424

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