Theorem opbrop 4702

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 4244	. . 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 464	. 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 4225	. . 3 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → ⟨𝐴, 𝐵⟩ ∈ V)
5		opexg 4225	. . 3 ⊢ ((𝐶 ∈ 𝑆 ∧ 𝐷 ∈ 𝑆) → ⟨𝐶, 𝐷⟩ ∈ V)
6		eleq1 2240	. . . . . 6 ⊢ (𝑥 = ⟨𝐴, 𝐵⟩ → (𝑥 ∈ (𝑆 × 𝑆) ↔ ⟨𝐴, 𝐵⟩ ∈ (𝑆 × 𝑆)))
7	6	anbi1d 465	. . . . 5 ⊢ (𝑥 = ⟨𝐴, 𝐵⟩ → ((𝑥 ∈ (𝑆 × 𝑆) ∧ 𝑦 ∈ (𝑆 × 𝑆)) ↔ (⟨𝐴, 𝐵⟩ ∈ (𝑆 × 𝑆) ∧ 𝑦 ∈ (𝑆 × 𝑆))))
8		eqeq1 2184	. . . . . . . 8 ⊢ (𝑥 = ⟨𝐴, 𝐵⟩ → (𝑥 = ⟨𝑧, 𝑤⟩ ↔ ⟨𝐴, 𝐵⟩ = ⟨𝑧, 𝑤⟩))
9	8	anbi1d 465	. . . . . . 7 ⊢ (𝑥 = ⟨𝐴, 𝐵⟩ → ((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ↔ (⟨𝐴, 𝐵⟩ = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩)))
10	9	anbi1d 465	. . . . . 6 ⊢ (𝑥 = ⟨𝐴, 𝐵⟩ → (((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ 𝜑) ↔ ((⟨𝐴, 𝐵⟩ = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ 𝜑)))
11	10	4exbidv 1870	. . . . 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 473	. . . 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 2240	. . . . . 6 ⊢ (𝑦 = ⟨𝐶, 𝐷⟩ → (𝑦 ∈ (𝑆 × 𝑆) ↔ ⟨𝐶, 𝐷⟩ ∈ (𝑆 × 𝑆)))
14	13	anbi2d 464	. . . . 5 ⊢ (𝑦 = ⟨𝐶, 𝐷⟩ → ((⟨𝐴, 𝐵⟩ ∈ (𝑆 × 𝑆) ∧ 𝑦 ∈ (𝑆 × 𝑆)) ↔ (⟨𝐴, 𝐵⟩ ∈ (𝑆 × 𝑆) ∧ ⟨𝐶, 𝐷⟩ ∈ (𝑆 × 𝑆))))
15		eqeq1 2184	. . . . . . . 8 ⊢ (𝑦 = ⟨𝐶, 𝐷⟩ → (𝑦 = ⟨𝑣, 𝑢⟩ ↔ ⟨𝐶, 𝐷⟩ = ⟨𝑣, 𝑢⟩))
16	15	anbi2d 464	. . . . . . 7 ⊢ (𝑦 = ⟨𝐶, 𝐷⟩ → ((⟨𝐴, 𝐵⟩ = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ↔ (⟨𝐴, 𝐵⟩ = ⟨𝑧, 𝑤⟩ ∧ ⟨𝐶, 𝐷⟩ = ⟨𝑣, 𝑢⟩)))
17	16	anbi1d 465	. . . . . 6 ⊢ (𝑦 = ⟨𝐶, 𝐷⟩ → (((⟨𝐴, 𝐵⟩ = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ 𝜑) ↔ ((⟨𝐴, 𝐵⟩ = ⟨𝑧, 𝑤⟩ ∧ ⟨𝐶, 𝐷⟩ = ⟨𝑣, 𝑢⟩) ∧ 𝜑)))
18	17	4exbidv 1870	. . . . 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 473	. . . 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 4266	. . 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 289	. 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		opelxpi 4655	. . . 4 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → ⟨𝐴, 𝐵⟩ ∈ (𝑆 × 𝑆))
24		opelxpi 4655	. . . 4 ⊢ ((𝐶 ∈ 𝑆 ∧ 𝐷 ∈ 𝑆) → ⟨𝐶, 𝐷⟩ ∈ (𝑆 × 𝑆))
25	23, 24	anim12i 338	. . 3 ⊢ (((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) ∧ (𝐶 ∈ 𝑆 ∧ 𝐷 ∈ 𝑆)) → (⟨𝐴, 𝐵⟩ ∈ (𝑆 × 𝑆) ∧ ⟨𝐶, 𝐷⟩ ∈ (𝑆 × 𝑆)))
26	25	biantrurd 305	. 2 ⊢ (((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) ∧ (𝐶 ∈ 𝑆 ∧ 𝐷 ∈ 𝑆)) → (𝜓 ↔ ((⟨𝐴, 𝐵⟩ ∈ (𝑆 × 𝑆) ∧ ⟨𝐶, 𝐷⟩ ∈ (𝑆 × 𝑆)) ∧ 𝜓)))
27	3, 22, 26	3bitr4d 220	1 ⊢ (((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) ∧ (𝐶 ∈ 𝑆 ∧ 𝐷 ∈ 𝑆)) → (⟨𝐴, 𝐵⟩𝑅⟨𝐶, 𝐷⟩ ↔ 𝜓))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1353  ∃wex 1492   ∈ wcel 2148  Vcvv 2737  ⟨cop 3594   class class class wbr 4000  {copab 4060   × cxp 4621

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171  ax-pr 4206

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-br 4001  df-opab 4062  df-xp 4629

This theorem is referenced by:  ecopoveq  6624  oviec  6635