Theorem opbrop 4842

Description: Ordered pair membership in a relation. Special case. (Contributed by NM, 5-Aug-1995.)

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 groups:   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

Allowed substitution hints:   φ(z,w,v,u)   ψ(x,y)   R(x,y,z,w,v,u)

Proof of Theorem opbrop

Step	Hyp	Ref	Expression
1		opbrop.1	. . . 4 ⊢ (((z = A ∧ w = B) ∧ (v = C ∧ u = D)) → (φ ↔ ψ))
2	1	copsex4g 4611	. . 3 ⊢ (((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 684	. 2 ⊢ (((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		opexg 4588	. . 3 ⊢ ((A ∈ S ∧ B ∈ S) → ⟨A, B⟩ ∈ V)
5		opexg 4588	. . 3 ⊢ ((C ∈ S ∧ D ∈ S) → ⟨C, D⟩ ∈ V)
6		eleq1 2413	. . . . . 6 ⊢ (x = ⟨A, B⟩ → (x ∈ (S × S) ↔ ⟨A, B⟩ ∈ (S × S)))
7	6	anbi1d 685	. . . . 5 ⊢ (x = ⟨A, B⟩ → ((x ∈ (S × S) ∧ y ∈ (S × S)) ↔ (⟨A, B⟩ ∈ (S × S) ∧ y ∈ (S × S))))
8		eqeq1 2359	. . . . . . . 8 ⊢ (x = ⟨A, B⟩ → (x = ⟨z, w⟩ ↔ ⟨A, B⟩ = ⟨z, w⟩))
9	8	anbi1d 685	. . . . . . 7 ⊢ (x = ⟨A, B⟩ → ((x = ⟨z, w⟩ ∧ y = ⟨v, u⟩) ↔ (⟨A, B⟩ = ⟨z, w⟩ ∧ y = ⟨v, u⟩)))
10	9	anbi1d 685	. . . . . 6 ⊢ (x = ⟨A, B⟩ → (((x = ⟨z, w⟩ ∧ y = ⟨v, u⟩) ∧ φ) ↔ ((⟨A, B⟩ = ⟨z, w⟩ ∧ y = ⟨v, u⟩) ∧ φ)))
11	10	4exbidv 1630	. . . . 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 691	. . . 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 2413	. . . . . 6 ⊢ (y = ⟨C, D⟩ → (y ∈ (S × S) ↔ ⟨C, D⟩ ∈ (S × S)))
14	13	anbi2d 684	. . . . 5 ⊢ (y = ⟨C, D⟩ → ((⟨A, B⟩ ∈ (S × S) ∧ y ∈ (S × S)) ↔ (⟨A, B⟩ ∈ (S × S) ∧ ⟨C, D⟩ ∈ (S × S))))
15		eqeq1 2359	. . . . . . . 8 ⊢ (y = ⟨C, D⟩ → (y = ⟨v, u⟩ ↔ ⟨C, D⟩ = ⟨v, u⟩))
16	15	anbi2d 684	. . . . . . 7 ⊢ (y = ⟨C, D⟩ → ((⟨A, B⟩ = ⟨z, w⟩ ∧ y = ⟨v, u⟩) ↔ (⟨A, B⟩ = ⟨z, w⟩ ∧ ⟨C, D⟩ = ⟨v, u⟩)))
17	16	anbi1d 685	. . . . . 6 ⊢ (y = ⟨C, D⟩ → (((⟨A, B⟩ = ⟨z, w⟩ ∧ y = ⟨v, u⟩) ∧ φ) ↔ ((⟨A, B⟩ = ⟨z, w⟩ ∧ ⟨C, D⟩ = ⟨v, u⟩) ∧ φ)))
18	17	4exbidv 1630	. . . . 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 691	. . . 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	12, 19, 20	brabg 4707	. . 3 ⊢ ((⟨A, B⟩ ∈ V ∧ ⟨C, D⟩ ∈ V) → (⟨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	4, 5, 21	syl2an 463	. 2 ⊢ (((A ∈ S ∧ B ∈ S) ∧ (C ∈ S ∧ D ∈ S)) → (⟨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))))
23		opelxp 4812	. . . . 5 ⊢ (⟨A, B⟩ ∈ (S × S) ↔ (A ∈ S ∧ B ∈ S))
24		opelxp 4812	. . . . 5 ⊢ (⟨C, D⟩ ∈ (S × S) ↔ (C ∈ S ∧ D ∈ S))
25	23, 24	anbi12i 678	. . . 4 ⊢ ((⟨A, B⟩ ∈ (S × S) ∧ ⟨C, D⟩ ∈ (S × S)) ↔ ((A ∈ S ∧ B ∈ S) ∧ (C ∈ S ∧ D ∈ S)))
26	25	biimpri 197	. . 3 ⊢ (((A ∈ S ∧ B ∈ S) ∧ (C ∈ S ∧ D ∈ S)) → (⟨A, B⟩ ∈ (S × S) ∧ ⟨C, D⟩ ∈ (S × S)))
27	26	biantrurd 494	. 2 ⊢ (((A ∈ S ∧ B ∈ S) ∧ (C ∈ S ∧ D ∈ S)) → (ψ ↔ ((⟨A, B⟩ ∈ (S × S) ∧ ⟨C, D⟩ ∈ (S × S)) ∧ ψ)))
28	3, 22, 27	3bitr4d 276	1 ⊢ (((A ∈ S ∧ B ∈ S) ∧ (C ∈ S ∧ D ∈ S)) → (⟨A, B⟩R⟨C, D⟩ ↔ ψ))

Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358  ∃wex 1541   = wceq 1642   ∈ wcel 1710  Vcvv 2860  ⟨cop 4562  {copab 4623   class class class wbr 4640   × cxp 4771

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4079  ax-xp 4080  ax-cnv 4081  ax-1c 4082  ax-sset 4083  ax-si 4084  ax-ins2 4085  ax-ins3 4086  ax-typlower 4087  ax-sn 4088

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-reu 2622  df-rmo 2623  df-rab 2624  df-v 2862  df-sbc 3048  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-symdif 3217  df-ss 3260  df-pss 3262  df-nul 3552  df-if 3664  df-pw 3725  df-sn 3742  df-pr 3743  df-uni 3893  df-int 3928  df-opk 4059  df-1c 4137  df-pw1 4138  df-uni1 4139  df-xpk 4186  df-cnvk 4187  df-ins2k 4188  df-ins3k 4189  df-imak 4190  df-cok 4191  df-p6 4192  df-sik 4193  df-ssetk 4194  df-imagek 4195  df-idk 4196  df-iota 4340  df-0c 4378  df-addc 4379  df-nnc 4380  df-fin 4381  df-lefin 4441  df-ltfin 4442  df-ncfin 4443  df-tfin 4444  df-evenfin 4445  df-oddfin 4446  df-sfin 4447  df-spfin 4448  df-phi 4566  df-op 4567  df-proj1 4568  df-proj2 4569  df-opab 4624  df-br 4641  df-xp 4785

This theorem is referenced by: (None)