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Theorem rbropapd 5430
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 5048 . . . 4 (𝐹𝑀𝑃 ↔ ⟨𝐹, 𝑃⟩ ∈ 𝑀)
2 rbropapd.1 . . . . 5 (𝜑𝑀 = {⟨𝑓, 𝑝⟩ ∣ (𝑓𝑊𝑝𝜓)})
32eleq2d 2901 . . . 4 (𝜑 → (⟨𝐹, 𝑃⟩ ∈ 𝑀 ↔ ⟨𝐹, 𝑃⟩ ∈ {⟨𝑓, 𝑝⟩ ∣ (𝑓𝑊𝑝𝜓)}))
41, 3syl5bb 286 . . 3 (𝜑 → (𝐹𝑀𝑃 ↔ ⟨𝐹, 𝑃⟩ ∈ {⟨𝑓, 𝑝⟩ ∣ (𝑓𝑊𝑝𝜓)}))
5 breq12 5052 . . . . 5 ((𝑓 = 𝐹𝑝 = 𝑃) → (𝑓𝑊𝑝𝐹𝑊𝑃))
6 rbropapd.2 . . . . 5 ((𝑓 = 𝐹𝑝 = 𝑃) → (𝜓𝜒))
75, 6anbi12d 633 . . . 4 ((𝑓 = 𝐹𝑝 = 𝑃) → ((𝑓𝑊𝑝𝜓) ↔ (𝐹𝑊𝑃𝜒)))
87opelopabga 5401 . . 3 ((𝐹𝑋𝑃𝑌) → (⟨𝐹, 𝑃⟩ ∈ {⟨𝑓, 𝑝⟩ ∣ (𝑓𝑊𝑝𝜓)} ↔ (𝐹𝑊𝑃𝜒)))
94, 8sylan9bb 513 . 2 ((𝜑 ∧ (𝐹𝑋𝑃𝑌)) → (𝐹𝑀𝑃 ↔ (𝐹𝑊𝑃𝜒)))
109ex 416 1 (𝜑 → ((𝐹𝑋𝑃𝑌) → (𝐹𝑀𝑃 ↔ (𝐹𝑊𝑃𝜒))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1538  wcel 2115  cop 4554   class class class wbr 5047  {copab 5109
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5184  ax-nul 5191  ax-pr 5311
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-rab 3141  df-v 3481  df-dif 3921  df-un 3923  df-in 3925  df-ss 3935  df-nul 4275  df-if 4449  df-sn 4549  df-pr 4551  df-op 4555  df-br 5048  df-opab 5110
This theorem is referenced by:  rbropap  5431
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