Theorem inxprnres 38252

Description: Restriction of a class as a class of ordered pairs. (Contributed by Peter Mazsa, 2-Jan-2019.)

Assertion

Ref	Expression
inxprnres	⊢ (𝑅 ∩ (𝐴 × ran (𝑅 ↾ 𝐴))) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑦)}

Distinct variable groups:   𝑥,𝐴,𝑦   𝑥,𝑅,𝑦

Proof of Theorem inxprnres

Dummy variables 𝑤 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		relinxp 5804	. 2 ⊢ Rel (𝑅 ∩ (𝐴 × ran (𝑅 ↾ 𝐴)))
2		relopabv 5811	. 2 ⊢ Rel {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑦)}
3		eleq1w 2816	. . . . . 6 ⊢ (𝑥 = 𝑧 → (𝑥 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴))
4		breq1 5126	. . . . . 6 ⊢ (𝑥 = 𝑧 → (𝑥𝑅𝑦 ↔ 𝑧𝑅𝑦))
5	3, 4	anbi12d 632	. . . . 5 ⊢ (𝑥 = 𝑧 → ((𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑦) ↔ (𝑧 ∈ 𝐴 ∧ 𝑧𝑅𝑦)))
6		breq2 5127	. . . . . 6 ⊢ (𝑦 = 𝑤 → (𝑧𝑅𝑦 ↔ 𝑧𝑅𝑤))
7	6	anbi2d 630	. . . . 5 ⊢ (𝑦 = 𝑤 → ((𝑧 ∈ 𝐴 ∧ 𝑧𝑅𝑦) ↔ (𝑧 ∈ 𝐴 ∧ 𝑧𝑅𝑤)))
8	5, 7	opelopabg 5523	. . . 4 ⊢ ((𝑧 ∈ V ∧ 𝑤 ∈ V) → (⟨𝑧, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑦)} ↔ (𝑧 ∈ 𝐴 ∧ 𝑧𝑅𝑤)))
9	8	el2v 3470	. . 3 ⊢ (⟨𝑧, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑦)} ↔ (𝑧 ∈ 𝐴 ∧ 𝑧𝑅𝑤))
10		brinxprnres 38251	. . . 4 ⊢ (𝑤 ∈ V → (𝑧(𝑅 ∩ (𝐴 × ran (𝑅 ↾ 𝐴)))𝑤 ↔ (𝑧 ∈ 𝐴 ∧ 𝑧𝑅𝑤)))
11	10	elv 3468	. . 3 ⊢ (𝑧(𝑅 ∩ (𝐴 × ran (𝑅 ↾ 𝐴)))𝑤 ↔ (𝑧 ∈ 𝐴 ∧ 𝑧𝑅𝑤))
12		df-br 5124	. . 3 ⊢ (𝑧(𝑅 ∩ (𝐴 × ran (𝑅 ↾ 𝐴)))𝑤 ↔ ⟨𝑧, 𝑤⟩ ∈ (𝑅 ∩ (𝐴 × ran (𝑅 ↾ 𝐴))))
13	9, 11, 12	3bitr2ri 300	. 2 ⊢ (⟨𝑧, 𝑤⟩ ∈ (𝑅 ∩ (𝐴 × ran (𝑅 ↾ 𝐴))) ↔ ⟨𝑧, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑦)})
14	1, 2, 13	eqrelriiv 5780	1 ⊢ (𝑅 ∩ (𝐴 × ran (𝑅 ↾ 𝐴))) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑦)}

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1539   ∈ wcel 2107  Vcvv 3463   ∩ cin 3930  ⟨cop 4612   class class class wbr 5123  {copab 5185   × cxp 5663  ran crn 5666   ↾ cres 5667

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2706  ax-sep 5276  ax-nul 5286  ax-pr 5412

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2713  df-cleq 2726  df-clel 2808  df-ral 3051  df-rex 3060  df-rab 3420  df-v 3465  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-sn 4607  df-pr 4609  df-op 4613  df-br 5124  df-opab 5186  df-xp 5671  df-rel 5672  df-cnv 5673  df-dm 5675  df-rn 5676  df-res 5677

This theorem is referenced by:  dfres4  38253