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Theorem rbropapd 5574
Description: Properties of a pair in an extended binary relation. (Contributed by Alexander van der Vekens, 30-Oct-2017.)
Hypotheses
Ref Expression
rbropapd.1 (𝜑𝑀 = {⟨𝑓, 𝑝⟩ ∣ (𝑓𝑊𝑝𝜓)})
rbropapd.2 ((𝑓 = 𝐹𝑝 = 𝑃) → (𝜓𝜒))
Assertion
Ref Expression
rbropapd (𝜑 → ((𝐹𝑋𝑃𝑌) → (𝐹𝑀𝑃 ↔ (𝐹𝑊𝑃𝜒))))
Distinct variable groups:   𝑓,𝐹,𝑝   𝑃,𝑓,𝑝   𝑓,𝑊,𝑝   𝜒,𝑓,𝑝
Allowed substitution hints:   𝜑(𝑓,𝑝)   𝜓(𝑓,𝑝)   𝑀(𝑓,𝑝)   𝑋(𝑓,𝑝)   𝑌(𝑓,𝑝)

Proof of Theorem rbropapd
StepHypRef Expression
1 df-br 5149 . . . 4 (𝐹𝑀𝑃 ↔ ⟨𝐹, 𝑃⟩ ∈ 𝑀)
2 rbropapd.1 . . . . 5 (𝜑𝑀 = {⟨𝑓, 𝑝⟩ ∣ (𝑓𝑊𝑝𝜓)})
32eleq2d 2825 . . . 4 (𝜑 → (⟨𝐹, 𝑃⟩ ∈ 𝑀 ↔ ⟨𝐹, 𝑃⟩ ∈ {⟨𝑓, 𝑝⟩ ∣ (𝑓𝑊𝑝𝜓)}))
41, 3bitrid 283 . . 3 (𝜑 → (𝐹𝑀𝑃 ↔ ⟨𝐹, 𝑃⟩ ∈ {⟨𝑓, 𝑝⟩ ∣ (𝑓𝑊𝑝𝜓)}))
5 breq12 5153 . . . . 5 ((𝑓 = 𝐹𝑝 = 𝑃) → (𝑓𝑊𝑝𝐹𝑊𝑃))
6 rbropapd.2 . . . . 5 ((𝑓 = 𝐹𝑝 = 𝑃) → (𝜓𝜒))
75, 6anbi12d 632 . . . 4 ((𝑓 = 𝐹𝑝 = 𝑃) → ((𝑓𝑊𝑝𝜓) ↔ (𝐹𝑊𝑃𝜒)))
87opelopabga 5543 . . 3 ((𝐹𝑋𝑃𝑌) → (⟨𝐹, 𝑃⟩ ∈ {⟨𝑓, 𝑝⟩ ∣ (𝑓𝑊𝑝𝜓)} ↔ (𝐹𝑊𝑃𝜒)))
94, 8sylan9bb 509 . 2 ((𝜑 ∧ (𝐹𝑋𝑃𝑌)) → (𝐹𝑀𝑃 ↔ (𝐹𝑊𝑃𝜒)))
109ex 412 1 (𝜑 → ((𝐹𝑋𝑃𝑌) → (𝐹𝑀𝑃 ↔ (𝐹𝑊𝑃𝜒))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1537  wcel 2106  cop 4637   class class class wbr 5148  {copab 5210
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211
This theorem is referenced by:  rbropap  5575
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