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Theorem opbrop 4588
Description: Ordered pair membership in a relation. Special case. (Contributed by NM, 5-Aug-1995.)
Hypotheses
Ref Expression
opbrop.1 (((𝑧 = 𝐴𝑤 = 𝐵) ∧ (𝑣 = 𝐶𝑢 = 𝐷)) → (𝜑𝜓))
opbrop.2 𝑅 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (𝑆 × 𝑆) ∧ 𝑦 ∈ (𝑆 × 𝑆)) ∧ ∃𝑧𝑤𝑣𝑢((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ 𝜑))}
Assertion
Ref Expression
opbrop (((𝐴𝑆𝐵𝑆) ∧ (𝐶𝑆𝐷𝑆)) → (⟨𝐴, 𝐵𝑅𝐶, 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
StepHypRef Expression
1 opbrop.1 . . . 4 (((𝑧 = 𝐴𝑤 = 𝐵) ∧ (𝑣 = 𝐶𝑢 = 𝐷)) → (𝜑𝜓))
21copsex4g 4139 . . 3 (((𝐴𝑆𝐵𝑆) ∧ (𝐶𝑆𝐷𝑆)) → (∃𝑧𝑤𝑣𝑢((⟨𝐴, 𝐵⟩ = ⟨𝑧, 𝑤⟩ ∧ ⟨𝐶, 𝐷⟩ = ⟨𝑣, 𝑢⟩) ∧ 𝜑) ↔ 𝜓))
32anbi2d 459 . 2 (((𝐴𝑆𝐵𝑆) ∧ (𝐶𝑆𝐷𝑆)) → (((⟨𝐴, 𝐵⟩ ∈ (𝑆 × 𝑆) ∧ ⟨𝐶, 𝐷⟩ ∈ (𝑆 × 𝑆)) ∧ ∃𝑧𝑤𝑣𝑢((⟨𝐴, 𝐵⟩ = ⟨𝑧, 𝑤⟩ ∧ ⟨𝐶, 𝐷⟩ = ⟨𝑣, 𝑢⟩) ∧ 𝜑)) ↔ ((⟨𝐴, 𝐵⟩ ∈ (𝑆 × 𝑆) ∧ ⟨𝐶, 𝐷⟩ ∈ (𝑆 × 𝑆)) ∧ 𝜓)))
4 opexg 4120 . . 3 ((𝐴𝑆𝐵𝑆) → ⟨𝐴, 𝐵⟩ ∈ V)
5 opexg 4120 . . 3 ((𝐶𝑆𝐷𝑆) → ⟨𝐶, 𝐷⟩ ∈ V)
6 eleq1 2180 . . . . . 6 (𝑥 = ⟨𝐴, 𝐵⟩ → (𝑥 ∈ (𝑆 × 𝑆) ↔ ⟨𝐴, 𝐵⟩ ∈ (𝑆 × 𝑆)))
76anbi1d 460 . . . . 5 (𝑥 = ⟨𝐴, 𝐵⟩ → ((𝑥 ∈ (𝑆 × 𝑆) ∧ 𝑦 ∈ (𝑆 × 𝑆)) ↔ (⟨𝐴, 𝐵⟩ ∈ (𝑆 × 𝑆) ∧ 𝑦 ∈ (𝑆 × 𝑆))))
8 eqeq1 2124 . . . . . . . 8 (𝑥 = ⟨𝐴, 𝐵⟩ → (𝑥 = ⟨𝑧, 𝑤⟩ ↔ ⟨𝐴, 𝐵⟩ = ⟨𝑧, 𝑤⟩))
98anbi1d 460 . . . . . . 7 (𝑥 = ⟨𝐴, 𝐵⟩ → ((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ↔ (⟨𝐴, 𝐵⟩ = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩)))
109anbi1d 460 . . . . . 6 (𝑥 = ⟨𝐴, 𝐵⟩ → (((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ 𝜑) ↔ ((⟨𝐴, 𝐵⟩ = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ 𝜑)))
11104exbidv 1826 . . . . 5 (𝑥 = ⟨𝐴, 𝐵⟩ → (∃𝑧𝑤𝑣𝑢((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ 𝜑) ↔ ∃𝑧𝑤𝑣𝑢((⟨𝐴, 𝐵⟩ = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ 𝜑)))
127, 11anbi12d 464 . . . 4 (𝑥 = ⟨𝐴, 𝐵⟩ → (((𝑥 ∈ (𝑆 × 𝑆) ∧ 𝑦 ∈ (𝑆 × 𝑆)) ∧ ∃𝑧𝑤𝑣𝑢((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ 𝜑)) ↔ ((⟨𝐴, 𝐵⟩ ∈ (𝑆 × 𝑆) ∧ 𝑦 ∈ (𝑆 × 𝑆)) ∧ ∃𝑧𝑤𝑣𝑢((⟨𝐴, 𝐵⟩ = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ 𝜑))))
13 eleq1 2180 . . . . . 6 (𝑦 = ⟨𝐶, 𝐷⟩ → (𝑦 ∈ (𝑆 × 𝑆) ↔ ⟨𝐶, 𝐷⟩ ∈ (𝑆 × 𝑆)))
1413anbi2d 459 . . . . 5 (𝑦 = ⟨𝐶, 𝐷⟩ → ((⟨𝐴, 𝐵⟩ ∈ (𝑆 × 𝑆) ∧ 𝑦 ∈ (𝑆 × 𝑆)) ↔ (⟨𝐴, 𝐵⟩ ∈ (𝑆 × 𝑆) ∧ ⟨𝐶, 𝐷⟩ ∈ (𝑆 × 𝑆))))
15 eqeq1 2124 . . . . . . . 8 (𝑦 = ⟨𝐶, 𝐷⟩ → (𝑦 = ⟨𝑣, 𝑢⟩ ↔ ⟨𝐶, 𝐷⟩ = ⟨𝑣, 𝑢⟩))
1615anbi2d 459 . . . . . . 7 (𝑦 = ⟨𝐶, 𝐷⟩ → ((⟨𝐴, 𝐵⟩ = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ↔ (⟨𝐴, 𝐵⟩ = ⟨𝑧, 𝑤⟩ ∧ ⟨𝐶, 𝐷⟩ = ⟨𝑣, 𝑢⟩)))
1716anbi1d 460 . . . . . 6 (𝑦 = ⟨𝐶, 𝐷⟩ → (((⟨𝐴, 𝐵⟩ = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ 𝜑) ↔ ((⟨𝐴, 𝐵⟩ = ⟨𝑧, 𝑤⟩ ∧ ⟨𝐶, 𝐷⟩ = ⟨𝑣, 𝑢⟩) ∧ 𝜑)))
18174exbidv 1826 . . . . 5 (𝑦 = ⟨𝐶, 𝐷⟩ → (∃𝑧𝑤𝑣𝑢((⟨𝐴, 𝐵⟩ = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ 𝜑) ↔ ∃𝑧𝑤𝑣𝑢((⟨𝐴, 𝐵⟩ = ⟨𝑧, 𝑤⟩ ∧ ⟨𝐶, 𝐷⟩ = ⟨𝑣, 𝑢⟩) ∧ 𝜑)))
1914, 18anbi12d 464 . . . 4 (𝑦 = ⟨𝐶, 𝐷⟩ → (((⟨𝐴, 𝐵⟩ ∈ (𝑆 × 𝑆) ∧ 𝑦 ∈ (𝑆 × 𝑆)) ∧ ∃𝑧𝑤𝑣𝑢((⟨𝐴, 𝐵⟩ = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ 𝜑)) ↔ ((⟨𝐴, 𝐵⟩ ∈ (𝑆 × 𝑆) ∧ ⟨𝐶, 𝐷⟩ ∈ (𝑆 × 𝑆)) ∧ ∃𝑧𝑤𝑣𝑢((⟨𝐴, 𝐵⟩ = ⟨𝑧, 𝑤⟩ ∧ ⟨𝐶, 𝐷⟩ = ⟨𝑣, 𝑢⟩) ∧ 𝜑))))
20 opbrop.2 . . . 4 𝑅 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (𝑆 × 𝑆) ∧ 𝑦 ∈ (𝑆 × 𝑆)) ∧ ∃𝑧𝑤𝑣𝑢((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ 𝜑))}
2112, 19, 20brabg 4161 . . 3 ((⟨𝐴, 𝐵⟩ ∈ V ∧ ⟨𝐶, 𝐷⟩ ∈ V) → (⟨𝐴, 𝐵𝑅𝐶, 𝐷⟩ ↔ ((⟨𝐴, 𝐵⟩ ∈ (𝑆 × 𝑆) ∧ ⟨𝐶, 𝐷⟩ ∈ (𝑆 × 𝑆)) ∧ ∃𝑧𝑤𝑣𝑢((⟨𝐴, 𝐵⟩ = ⟨𝑧, 𝑤⟩ ∧ ⟨𝐶, 𝐷⟩ = ⟨𝑣, 𝑢⟩) ∧ 𝜑))))
224, 5, 21syl2an 287 . 2 (((𝐴𝑆𝐵𝑆) ∧ (𝐶𝑆𝐷𝑆)) → (⟨𝐴, 𝐵𝑅𝐶, 𝐷⟩ ↔ ((⟨𝐴, 𝐵⟩ ∈ (𝑆 × 𝑆) ∧ ⟨𝐶, 𝐷⟩ ∈ (𝑆 × 𝑆)) ∧ ∃𝑧𝑤𝑣𝑢((⟨𝐴, 𝐵⟩ = ⟨𝑧, 𝑤⟩ ∧ ⟨𝐶, 𝐷⟩ = ⟨𝑣, 𝑢⟩) ∧ 𝜑))))
23 opelxpi 4541 . . . 4 ((𝐴𝑆𝐵𝑆) → ⟨𝐴, 𝐵⟩ ∈ (𝑆 × 𝑆))
24 opelxpi 4541 . . . 4 ((𝐶𝑆𝐷𝑆) → ⟨𝐶, 𝐷⟩ ∈ (𝑆 × 𝑆))
2523, 24anim12i 336 . . 3 (((𝐴𝑆𝐵𝑆) ∧ (𝐶𝑆𝐷𝑆)) → (⟨𝐴, 𝐵⟩ ∈ (𝑆 × 𝑆) ∧ ⟨𝐶, 𝐷⟩ ∈ (𝑆 × 𝑆)))
2625biantrurd 303 . 2 (((𝐴𝑆𝐵𝑆) ∧ (𝐶𝑆𝐷𝑆)) → (𝜓 ↔ ((⟨𝐴, 𝐵⟩ ∈ (𝑆 × 𝑆) ∧ ⟨𝐶, 𝐷⟩ ∈ (𝑆 × 𝑆)) ∧ 𝜓)))
273, 22, 263bitr4d 219 1 (((𝐴𝑆𝐵𝑆) ∧ (𝐶𝑆𝐷𝑆)) → (⟨𝐴, 𝐵𝑅𝐶, 𝐷⟩ ↔ 𝜓))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104   = wceq 1316  wex 1453  wcel 1465  Vcvv 2660  cop 3500   class class class wbr 3899  {copab 3958   × cxp 4507
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-v 2662  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-br 3900  df-opab 3960  df-xp 4515
This theorem is referenced by:  ecopoveq  6492  oviec  6503
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