Theorem opbrop 5764

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		opelxpi 5703	. . 3 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → ⟨𝐴, 𝐵⟩ ∈ (𝑆 × 𝑆))
2		opelxpi 5703	. . 3 ⊢ ((𝐶 ∈ 𝑆 ∧ 𝐷 ∈ 𝑆) → ⟨𝐶, 𝐷⟩ ∈ (𝑆 × 𝑆))
3	1, 2	anim12i 625	. 2 ⊢ (((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) ∧ (𝐶 ∈ 𝑆 ∧ 𝐷 ∈ 𝑆)) → (⟨𝐴, 𝐵⟩ ∈ (𝑆 × 𝑆) ∧ ⟨𝐶, 𝐷⟩ ∈ (𝑆 × 𝑆)))
4		opex 5450	. . . 4 ⊢ ⟨𝐴, 𝐵⟩ ∈ V
5		opex 5450	. . . 4 ⊢ ⟨𝐶, 𝐷⟩ ∈ V
6		eleq1 2854	. . . . . 6 ⊢ (𝑥 = ⟨𝐴, 𝐵⟩ → (𝑥 ∈ (𝑆 × 𝑆) ↔ ⟨𝐴, 𝐵⟩ ∈ (𝑆 × 𝑆)))
7	6	anbi1d 643	. . . . 5 ⊢ (𝑥 = ⟨𝐴, 𝐵⟩ → ((𝑥 ∈ (𝑆 × 𝑆) ∧ 𝑦 ∈ (𝑆 × 𝑆)) ↔ (⟨𝐴, 𝐵⟩ ∈ (𝑆 × 𝑆) ∧ 𝑦 ∈ (𝑆 × 𝑆))))
8		eqeq1 2770	. . . . . . . 8 ⊢ (𝑥 = ⟨𝐴, 𝐵⟩ → (𝑥 = ⟨𝑧, 𝑤⟩ ↔ ⟨𝐴, 𝐵⟩ = ⟨𝑧, 𝑤⟩))
9	8	anbi1d 643	. . . . . . 7 ⊢ (𝑥 = ⟨𝐴, 𝐵⟩ → ((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ↔ (⟨𝐴, 𝐵⟩ = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩)))
10	9	anbi1d 643	. . . . . 6 ⊢ (𝑥 = ⟨𝐴, 𝐵⟩ → (((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ 𝜑) ↔ ((⟨𝐴, 𝐵⟩ = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ 𝜑)))
11	10	4exbidv 1959	. . . . 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 644	. . . 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 2854	. . . . . 6 ⊢ (𝑦 = ⟨𝐶, 𝐷⟩ → (𝑦 ∈ (𝑆 × 𝑆) ↔ ⟨𝐶, 𝐷⟩ ∈ (𝑆 × 𝑆)))
14	13	anbi2d 642	. . . . 5 ⊢ (𝑦 = ⟨𝐶, 𝐷⟩ → ((⟨𝐴, 𝐵⟩ ∈ (𝑆 × 𝑆) ∧ 𝑦 ∈ (𝑆 × 𝑆)) ↔ (⟨𝐴, 𝐵⟩ ∈ (𝑆 × 𝑆) ∧ ⟨𝐶, 𝐷⟩ ∈ (𝑆 × 𝑆))))
15		eqeq1 2770	. . . . . . . 8 ⊢ (𝑦 = ⟨𝐶, 𝐷⟩ → (𝑦 = ⟨𝑣, 𝑢⟩ ↔ ⟨𝐶, 𝐷⟩ = ⟨𝑣, 𝑢⟩))
16	15	anbi2d 642	. . . . . . 7 ⊢ (𝑦 = ⟨𝐶, 𝐷⟩ → ((⟨𝐴, 𝐵⟩ = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ↔ (⟨𝐴, 𝐵⟩ = ⟨𝑧, 𝑤⟩ ∧ ⟨𝐶, 𝐷⟩ = ⟨𝑣, 𝑢⟩)))
17	16	anbi1d 643	. . . . . 6 ⊢ (𝑦 = ⟨𝐶, 𝐷⟩ → (((⟨𝐴, 𝐵⟩ = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ 𝜑) ↔ ((⟨𝐴, 𝐵⟩ = ⟨𝑧, 𝑤⟩ ∧ ⟨𝐶, 𝐷⟩ = ⟨𝑣, 𝑢⟩) ∧ 𝜑)))
18	17	4exbidv 1959	. . . . 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 644	. . . 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 5533	. . 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		opbrop.1	. . . . 5 ⊢ (((𝑧 = 𝐴 ∧ 𝑤 = 𝐵) ∧ (𝑣 = 𝐶 ∧ 𝑢 = 𝐷)) → (𝜑 ↔ 𝜓))
23	22	copsex4g 5483	. . . 4 ⊢ (((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) ∧ (𝐶 ∈ 𝑆 ∧ 𝐷 ∈ 𝑆)) → (∃𝑧∃𝑤∃𝑣∃𝑢((⟨𝐴, 𝐵⟩ = ⟨𝑧, 𝑤⟩ ∧ ⟨𝐶, 𝐷⟩ = ⟨𝑣, 𝑢⟩) ∧ 𝜑) ↔ 𝜓))
24	23	anbi2d 642	. . 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)))
25	21, 24	bitrid 286	. 2 ⊢ (((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) ∧ (𝐶 ∈ 𝑆 ∧ 𝐷 ∈ 𝑆)) → (⟨𝐴, 𝐵⟩𝑅⟨𝐶, 𝐷⟩ ↔ ((⟨𝐴, 𝐵⟩ ∈ (𝑆 × 𝑆) ∧ ⟨𝐶, 𝐷⟩ ∈ (𝑆 × 𝑆)) ∧ 𝜓)))
26	3, 25	mpbirand 720	1 ⊢ (((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) ∧ (𝐶 ∈ 𝑆 ∧ 𝐷 ∈ 𝑆)) → (⟨𝐴, 𝐵⟩𝑅⟨𝐶, 𝐷⟩ ↔ 𝜓))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570  ∃wex 1812   ∈ wcel 2146  ⟨cop 4600   class class class wbr 5114  {copab 5178   × cxp 5664

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-ext 2738  ax-sep 5262  ax-pr 5409

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-sb 2100  df-clab 2745  df-cleq 2758  df-clel 2841  df-ral 3083  df-rex 3093  df-rab 3420  df-v 3460  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4290  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-br 5115  df-opab 5179  df-xp 5672

This theorem is used by:  ecopoveq  8825