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Theorem 2rbropap 4931
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 1034 . . . . 5 ((𝑓𝑊𝑝𝜓𝜏) ↔ (𝑓𝑊𝑝 ∧ (𝜓𝜏)))
32opabbii 4643 . . . 4 {⟨𝑓, 𝑝⟩ ∣ (𝑓𝑊𝑝𝜓𝜏)} = {⟨𝑓, 𝑝⟩ ∣ (𝑓𝑊𝑝 ∧ (𝜓𝜏))}
41, 3syl6eq 2659 . . 3 (𝜑𝑀 = {⟨𝑓, 𝑝⟩ ∣ (𝑓𝑊𝑝 ∧ (𝜓𝜏))})
5 2rbropap.2 . . . 4 ((𝑓 = 𝐹𝑝 = 𝑃) → (𝜓𝜒))
6 2rbropap.3 . . . 4 ((𝑓 = 𝐹𝑝 = 𝑃) → (𝜏𝜃))
75, 6anbi12d 742 . . 3 ((𝑓 = 𝐹𝑝 = 𝑃) → ((𝜓𝜏) ↔ (𝜒𝜃)))
84, 7rbropap 4930 . 2 ((𝜑𝐹𝑋𝑃𝑌) → (𝐹𝑀𝑃 ↔ (𝐹𝑊𝑃 ∧ (𝜒𝜃))))
9 3anass 1034 . 2 ((𝐹𝑊𝑃𝜒𝜃) ↔ (𝐹𝑊𝑃 ∧ (𝜒𝜃)))
108, 9syl6bbr 276 1 ((𝜑𝐹𝑋𝑃𝑌) → (𝐹𝑀𝑃 ↔ (𝐹𝑊𝑃𝜒𝜃)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wa 382  w3a 1030   = wceq 1474  wcel 1976   class class class wbr 4577  {copab 4636
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-sep 4703  ax-nul 4712  ax-pr 4828
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-rab 2904  df-v 3174  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-sn 4125  df-pr 4127  df-op 4131  df-br 4578  df-opab 4638
This theorem is referenced by:  iswlkOn  40860  isPth  40924
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