Theorem rbropapd 6221

Description: Properties of a pair in an extended binary relation. (Contributed by Alexander van der Vekens, 30-Oct-2017.)

Hypotheses

Ref	Expression
rbropapd.1	⊢ (𝜑 → 𝑀 = {⟨𝑓, 𝑝⟩ ∣ (𝑓𝑊𝑝 ∧ 𝜓)})
rbropapd.2	⊢ ((𝑓 = 𝐹 ∧ 𝑝 = 𝑃) → (𝜓 ↔ 𝜒))

Assertion

Ref	Expression
rbropapd	⊢ (𝜑 → ((𝐹 ∈ 𝑋 ∧ 𝑃 ∈ 𝑌) → (𝐹𝑀𝑃 ↔ (𝐹𝑊𝑃 ∧ 𝜒))))

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

Step	Hyp	Ref	Expression
1		df-br 3990	. . . 4 ⊢ (𝐹𝑀𝑃 ↔ ⟨𝐹, 𝑃⟩ ∈ 𝑀)
2		rbropapd.1	. . . . 5 ⊢ (𝜑 → 𝑀 = {⟨𝑓, 𝑝⟩ ∣ (𝑓𝑊𝑝 ∧ 𝜓)})
3	2	eleq2d 2240	. . . 4 ⊢ (𝜑 → (⟨𝐹, 𝑃⟩ ∈ 𝑀 ↔ ⟨𝐹, 𝑃⟩ ∈ {⟨𝑓, 𝑝⟩ ∣ (𝑓𝑊𝑝 ∧ 𝜓)}))
4	1, 3	syl5bb 191	. . 3 ⊢ (𝜑 → (𝐹𝑀𝑃 ↔ ⟨𝐹, 𝑃⟩ ∈ {⟨𝑓, 𝑝⟩ ∣ (𝑓𝑊𝑝 ∧ 𝜓)}))
5		breq12 3994	. . . . 5 ⊢ ((𝑓 = 𝐹 ∧ 𝑝 = 𝑃) → (𝑓𝑊𝑝 ↔ 𝐹𝑊𝑃))
6		rbropapd.2	. . . . 5 ⊢ ((𝑓 = 𝐹 ∧ 𝑝 = 𝑃) → (𝜓 ↔ 𝜒))
7	5, 6	anbi12d 470	. . . 4 ⊢ ((𝑓 = 𝐹 ∧ 𝑝 = 𝑃) → ((𝑓𝑊𝑝 ∧ 𝜓) ↔ (𝐹𝑊𝑃 ∧ 𝜒)))
8	7	opelopabga 4248	. . 3 ⊢ ((𝐹 ∈ 𝑋 ∧ 𝑃 ∈ 𝑌) → (⟨𝐹, 𝑃⟩ ∈ {⟨𝑓, 𝑝⟩ ∣ (𝑓𝑊𝑝 ∧ 𝜓)} ↔ (𝐹𝑊𝑃 ∧ 𝜒)))
9	4, 8	sylan9bb 459	. 2 ⊢ ((𝜑 ∧ (𝐹 ∈ 𝑋 ∧ 𝑃 ∈ 𝑌)) → (𝐹𝑀𝑃 ↔ (𝐹𝑊𝑃 ∧ 𝜒)))
10	9	ex 114	1 ⊢ (𝜑 → ((𝐹 ∈ 𝑋 ∧ 𝑃 ∈ 𝑌) → (𝐹𝑀𝑃 ↔ (𝐹𝑊𝑃 ∧ 𝜒))))

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   = wceq 1348   ∈ wcel 2141  ⟨cop 3586   class class class wbr 3989  {copab 4049

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-br 3990  df-opab 4051

This theorem is referenced by:  rbropap  6222