Theorem 2rbropap 5444

Description: Properties of a pair in a restricted binary relation 𝑀 expressed as an ordered-pair class abstraction: 𝑀 is the binary relation 𝑊 restricted by the conditions 𝜓 and 𝜏. (Contributed by AV, 31-Jan-2021.)

Hypotheses

Ref	Expression
2rbropap.1	⊢ (𝜑 → 𝑀 = {⟨𝑓, 𝑝⟩ ∣ (𝑓𝑊𝑝 ∧ 𝜓 ∧ 𝜏)})
2rbropap.2	⊢ ((𝑓 = 𝐹 ∧ 𝑝 = 𝑃) → (𝜓 ↔ 𝜒))
2rbropap.3	⊢ ((𝑓 = 𝐹 ∧ 𝑝 = 𝑃) → (𝜏 ↔ 𝜃))

Assertion

Ref	Expression
2rbropap	⊢ ((𝜑 ∧ 𝐹 ∈ 𝑋 ∧ 𝑃 ∈ 𝑌) → (𝐹𝑀𝑃 ↔ (𝐹𝑊𝑃 ∧ 𝜒 ∧ 𝜃)))

Distinct variable groups:   𝑓,𝐹,𝑝   𝑃,𝑓,𝑝   𝑓,𝑊,𝑝   𝜒,𝑓,𝑝   𝜃,𝑓,𝑝

Allowed substitution hints:   𝜑(𝑓,𝑝)   𝜓(𝑓,𝑝)   𝜏(𝑓,𝑝)   𝑀(𝑓,𝑝)   𝑋(𝑓,𝑝)   𝑌(𝑓,𝑝)

Proof of Theorem 2rbropap

Step	Hyp	Ref	Expression
1		2rbropap.1	. . . 4 ⊢ (𝜑 → 𝑀 = {⟨𝑓, 𝑝⟩ ∣ (𝑓𝑊𝑝 ∧ 𝜓 ∧ 𝜏)})
2		3anass 1090	. . . . 5 ⊢ ((𝑓𝑊𝑝 ∧ 𝜓 ∧ 𝜏) ↔ (𝑓𝑊𝑝 ∧ (𝜓 ∧ 𝜏)))
3	2	opabbii 5126	. . . 4 ⊢ {⟨𝑓, 𝑝⟩ ∣ (𝑓𝑊𝑝 ∧ 𝜓 ∧ 𝜏)} = {⟨𝑓, 𝑝⟩ ∣ (𝑓𝑊𝑝 ∧ (𝜓 ∧ 𝜏))}
4	1, 3	syl6eq 2871	. . 3 ⊢ (𝜑 → 𝑀 = {⟨𝑓, 𝑝⟩ ∣ (𝑓𝑊𝑝 ∧ (𝜓 ∧ 𝜏))})
5		2rbropap.2	. . . 4 ⊢ ((𝑓 = 𝐹 ∧ 𝑝 = 𝑃) → (𝜓 ↔ 𝜒))
6		2rbropap.3	. . . 4 ⊢ ((𝑓 = 𝐹 ∧ 𝑝 = 𝑃) → (𝜏 ↔ 𝜃))
7	5, 6	anbi12d 632	. . 3 ⊢ ((𝑓 = 𝐹 ∧ 𝑝 = 𝑃) → ((𝜓 ∧ 𝜏) ↔ (𝜒 ∧ 𝜃)))
8	4, 7	rbropap 5443	. 2 ⊢ ((𝜑 ∧ 𝐹 ∈ 𝑋 ∧ 𝑃 ∈ 𝑌) → (𝐹𝑀𝑃 ↔ (𝐹𝑊𝑃 ∧ (𝜒 ∧ 𝜃))))
9		3anass 1090	. 2 ⊢ ((𝐹𝑊𝑃 ∧ 𝜒 ∧ 𝜃) ↔ (𝐹𝑊𝑃 ∧ (𝜒 ∧ 𝜃)))
10	8, 9	syl6bbr 291	1 ⊢ ((𝜑 ∧ 𝐹 ∈ 𝑋 ∧ 𝑃 ∈ 𝑌) → (𝐹𝑀𝑃 ↔ (𝐹𝑊𝑃 ∧ 𝜒 ∧ 𝜃)))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   ∧ w3a 1082   = wceq 1536   ∈ wcel 2113   class class class wbr 5059  {copab 5121

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2792  ax-sep 5196  ax-nul 5203  ax-pr 5323

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1084  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-mo 2621  df-eu 2653  df-clab 2799  df-cleq 2813  df-clel 2892  df-nfc 2962  df-rab 3146  df-v 3493  df-dif 3932  df-un 3934  df-in 3936  df-ss 3945  df-nul 4285  df-if 4461  df-sn 4561  df-pr 4563  df-op 4567  df-br 5060  df-opab 5122

This theorem is referenced by:  iswlkon  27437