Theorem rbropapd 6407

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 4089	. . . 4 ⊢ (𝐹𝑀𝑃 ↔ ⟨𝐹, 𝑃⟩ ∈ 𝑀)
2		rbropapd.1	. . . . 5 ⊢ (𝜑 → 𝑀 = {⟨𝑓, 𝑝⟩ ∣ (𝑓𝑊𝑝 ∧ 𝜓)})
3	2	eleq2d 2301	. . . 4 ⊢ (𝜑 → (⟨𝐹, 𝑃⟩ ∈ 𝑀 ↔ ⟨𝐹, 𝑃⟩ ∈ {⟨𝑓, 𝑝⟩ ∣ (𝑓𝑊𝑝 ∧ 𝜓)}))
4	1, 3	bitrid 192	. . 3 ⊢ (𝜑 → (𝐹𝑀𝑃 ↔ ⟨𝐹, 𝑃⟩ ∈ {⟨𝑓, 𝑝⟩ ∣ (𝑓𝑊𝑝 ∧ 𝜓)}))
5		breq12 4093	. . . . 5 ⊢ ((𝑓 = 𝐹 ∧ 𝑝 = 𝑃) → (𝑓𝑊𝑝 ↔ 𝐹𝑊𝑃))
6		rbropapd.2	. . . . 5 ⊢ ((𝑓 = 𝐹 ∧ 𝑝 = 𝑃) → (𝜓 ↔ 𝜒))
7	5, 6	anbi12d 473	. . . 4 ⊢ ((𝑓 = 𝐹 ∧ 𝑝 = 𝑃) → ((𝑓𝑊𝑝 ∧ 𝜓) ↔ (𝐹𝑊𝑃 ∧ 𝜒)))
8	7	opelopabga 4357	. . 3 ⊢ ((𝐹 ∈ 𝑋 ∧ 𝑃 ∈ 𝑌) → (⟨𝐹, 𝑃⟩ ∈ {⟨𝑓, 𝑝⟩ ∣ (𝑓𝑊𝑝 ∧ 𝜓)} ↔ (𝐹𝑊𝑃 ∧ 𝜒)))
9	4, 8	sylan9bb 462	. 2 ⊢ ((𝜑 ∧ (𝐹 ∈ 𝑋 ∧ 𝑃 ∈ 𝑌)) → (𝐹𝑀𝑃 ↔ (𝐹𝑊𝑃 ∧ 𝜒)))
10	9	ex 115	1 ⊢ (𝜑 → ((𝐹 ∈ 𝑋 ∧ 𝑃 ∈ 𝑌) → (𝐹𝑀𝑃 ↔ (𝐹𝑊𝑃 ∧ 𝜒))))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1397   ∈ wcel 2202  ⟨cop 3672   class class class wbr 4088  {copab 4149

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-v 2804  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-br 4089  df-opab 4151

This theorem is referenced by:  rbropap  6408