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Theorem 2rbropap 4941
 Description: Properties of a pair in a restricted binary relation 𝑀 expressed as an ordered-pair class abstraction: 𝑀 is the binary relation 𝑊 restricted by the conditions 𝜓 and 𝜏. (Contributed by AV, 31-Jan-2021.)
Hypotheses
Ref Expression
2rbropap.1 (𝜑𝑀 = {⟨𝑓, 𝑝⟩ ∣ (𝑓𝑊𝑝𝜓𝜏)})
2rbropap.2 ((𝑓 = 𝐹𝑝 = 𝑃) → (𝜓𝜒))
2rbropap.3 ((𝑓 = 𝐹𝑝 = 𝑃) → (𝜏𝜃))
Assertion
Ref Expression
2rbropap ((𝜑𝐹𝑋𝑃𝑌) → (𝐹𝑀𝑃 ↔ (𝐹𝑊𝑃𝜒𝜃)))
Distinct variable groups:   𝑓,𝐹,𝑝   𝑃,𝑓,𝑝   𝑓,𝑊,𝑝   𝜒,𝑓,𝑝   𝜃,𝑓,𝑝
Allowed substitution hints:   𝜑(𝑓,𝑝)   𝜓(𝑓,𝑝)   𝜏(𝑓,𝑝)   𝑀(𝑓,𝑝)   𝑋(𝑓,𝑝)   𝑌(𝑓,𝑝)

Proof of Theorem 2rbropap
StepHypRef Expression
1 2rbropap.1 . . . 4 (𝜑𝑀 = {⟨𝑓, 𝑝⟩ ∣ (𝑓𝑊𝑝𝜓𝜏)})
2 3anass 1035 . . . . 5 ((𝑓𝑊𝑝𝜓𝜏) ↔ (𝑓𝑊𝑝 ∧ (𝜓𝜏)))
32opabbii 4649 . . . 4 {⟨𝑓, 𝑝⟩ ∣ (𝑓𝑊𝑝𝜓𝜏)} = {⟨𝑓, 𝑝⟩ ∣ (𝑓𝑊𝑝 ∧ (𝜓𝜏))}
41, 3syl6eq 2660 . . 3 (𝜑𝑀 = {⟨𝑓, 𝑝⟩ ∣ (𝑓𝑊𝑝 ∧ (𝜓𝜏))})
5 2rbropap.2 . . . 4 ((𝑓 = 𝐹𝑝 = 𝑃) → (𝜓𝜒))
6 2rbropap.3 . . . 4 ((𝑓 = 𝐹𝑝 = 𝑃) → (𝜏𝜃))
75, 6anbi12d 743 . . 3 ((𝑓 = 𝐹𝑝 = 𝑃) → ((𝜓𝜏) ↔ (𝜒𝜃)))
84, 7rbropap 4940 . 2 ((𝜑𝐹𝑋𝑃𝑌) → (𝐹𝑀𝑃 ↔ (𝐹𝑊𝑃 ∧ (𝜒𝜃))))
9 3anass 1035 . 2 ((𝐹𝑊𝑃𝜒𝜃) ↔ (𝐹𝑊𝑃 ∧ (𝜒𝜃)))
108, 9syl6bbr 277 1 ((𝜑𝐹𝑋𝑃𝑌) → (𝐹𝑀𝑃 ↔ (𝐹𝑊𝑃𝜒𝜃)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   class class class wbr 4583  {copab 4642 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644 This theorem is referenced by:  iswlkOn  40865  isPth  40929
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