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Theorem 2rbropap 5303
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 1076 . . . . 5 ((𝑓𝑊𝑝𝜓𝜏) ↔ (𝑓𝑊𝑝 ∧ (𝜓𝜏)))
32opabbii 4996 . . . 4 {⟨𝑓, 𝑝⟩ ∣ (𝑓𝑊𝑝𝜓𝜏)} = {⟨𝑓, 𝑝⟩ ∣ (𝑓𝑊𝑝 ∧ (𝜓𝜏))}
41, 3syl6eq 2831 . . 3 (𝜑𝑀 = {⟨𝑓, 𝑝⟩ ∣ (𝑓𝑊𝑝 ∧ (𝜓𝜏))})
5 2rbropap.2 . . . 4 ((𝑓 = 𝐹𝑝 = 𝑃) → (𝜓𝜒))
6 2rbropap.3 . . . 4 ((𝑓 = 𝐹𝑝 = 𝑃) → (𝜏𝜃))
75, 6anbi12d 621 . . 3 ((𝑓 = 𝐹𝑝 = 𝑃) → ((𝜓𝜏) ↔ (𝜒𝜃)))
84, 7rbropap 5302 . 2 ((𝜑𝐹𝑋𝑃𝑌) → (𝐹𝑀𝑃 ↔ (𝐹𝑊𝑃 ∧ (𝜒𝜃))))
9 3anass 1076 . 2 ((𝐹𝑊𝑃𝜒𝜃) ↔ (𝐹𝑊𝑃 ∧ (𝜒𝜃)))
108, 9syl6bbr 281 1 ((𝜑𝐹𝑋𝑃𝑌) → (𝐹𝑀𝑃 ↔ (𝐹𝑊𝑃𝜒𝜃)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 387  w3a 1068   = wceq 1507  wcel 2050   class class class wbr 4929  {copab 4991
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2751  ax-sep 5060  ax-nul 5067  ax-pr 5186
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2760  df-cleq 2772  df-clel 2847  df-nfc 2919  df-rab 3098  df-v 3418  df-dif 3833  df-un 3835  df-in 3837  df-ss 3844  df-nul 4180  df-if 4351  df-sn 4442  df-pr 4444  df-op 4448  df-br 4930  df-opab 4992
This theorem is referenced by:  iswlkon  27141
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