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Theorem rbropapd 5557
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 5142 . . . 4 (𝐹𝑀𝑃 ↔ ⟨𝐹, 𝑃⟩ ∈ 𝑀)
2 rbropapd.1 . . . . 5 (𝜑𝑀 = {⟨𝑓, 𝑝⟩ ∣ (𝑓𝑊𝑝𝜓)})
32eleq2d 2813 . . . 4 (𝜑 → (⟨𝐹, 𝑃⟩ ∈ 𝑀 ↔ ⟨𝐹, 𝑃⟩ ∈ {⟨𝑓, 𝑝⟩ ∣ (𝑓𝑊𝑝𝜓)}))
41, 3bitrid 283 . . 3 (𝜑 → (𝐹𝑀𝑃 ↔ ⟨𝐹, 𝑃⟩ ∈ {⟨𝑓, 𝑝⟩ ∣ (𝑓𝑊𝑝𝜓)}))
5 breq12 5146 . . . . 5 ((𝑓 = 𝐹𝑝 = 𝑃) → (𝑓𝑊𝑝𝐹𝑊𝑃))
6 rbropapd.2 . . . . 5 ((𝑓 = 𝐹𝑝 = 𝑃) → (𝜓𝜒))
75, 6anbi12d 630 . . . 4 ((𝑓 = 𝐹𝑝 = 𝑃) → ((𝑓𝑊𝑝𝜓) ↔ (𝐹𝑊𝑃𝜒)))
87opelopabga 5526 . . 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 205  wa 395   = wceq 1533  wcel 2098  cop 4629   class class class wbr 5141  {copab 5203
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-br 5142  df-opab 5204
This theorem is referenced by:  rbropap  5558
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