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Theorem 2rbropap 5523
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 1095 . . . . 5 ((𝑓𝑊𝑝𝜓𝜏) ↔ (𝑓𝑊𝑝 ∧ (𝜓𝜏)))
32opabbii 5172 . . . 4 {⟨𝑓, 𝑝⟩ ∣ (𝑓𝑊𝑝𝜓𝜏)} = {⟨𝑓, 𝑝⟩ ∣ (𝑓𝑊𝑝 ∧ (𝜓𝜏))}
41, 3eqtrdi 2792 . . 3 (𝜑𝑀 = {⟨𝑓, 𝑝⟩ ∣ (𝑓𝑊𝑝 ∧ (𝜓𝜏))})
5 2rbropap.2 . . . 4 ((𝑓 = 𝐹𝑝 = 𝑃) → (𝜓𝜒))
6 2rbropap.3 . . . 4 ((𝑓 = 𝐹𝑝 = 𝑃) → (𝜏𝜃))
75, 6anbi12d 631 . . 3 ((𝑓 = 𝐹𝑝 = 𝑃) → ((𝜓𝜏) ↔ (𝜒𝜃)))
84, 7rbropap 5522 . 2 ((𝜑𝐹𝑋𝑃𝑌) → (𝐹𝑀𝑃 ↔ (𝐹𝑊𝑃 ∧ (𝜒𝜃))))
9 3anass 1095 . 2 ((𝐹𝑊𝑃𝜒𝜃) ↔ (𝐹𝑊𝑃 ∧ (𝜒𝜃)))
108, 9bitr4di 288 1 ((𝜑𝐹𝑋𝑃𝑌) → (𝐹𝑀𝑃 ↔ (𝐹𝑊𝑃𝜒𝜃)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  w3a 1087   = wceq 1541  wcel 2106   class class class wbr 5105  {copab 5167
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pr 5384
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2714  df-cleq 2728  df-clel 2814  df-rab 3408  df-v 3447  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-sn 4587  df-pr 4589  df-op 4593  df-br 5106  df-opab 5168
This theorem is referenced by:  iswlkon  28605
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