Theorem inxprnres 39047

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 5795	. 2 ⊢ Rel (𝑅 ∩ (𝐴 × ran (𝑅 ↾ 𝐴)))
2		relopabv 5802	. 2 ⊢ Rel {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑦)}
3		eleq1w 2843	. . . . . 6 ⊢ (𝑥 = 𝑧 → (𝑥 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴))
4		breq1 5106	. . . . . 6 ⊢ (𝑥 = 𝑧 → (𝑥𝑅𝑦 ↔ 𝑧𝑅𝑦))
5	3, 4	anbi12d 644	. . . . 5 ⊢ (𝑥 = 𝑧 → ((𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑦) ↔ (𝑧 ∈ 𝐴 ∧ 𝑧𝑅𝑦)))
6		breq2 5107	. . . . . 6 ⊢ (𝑦 = 𝑤 → (𝑧𝑅𝑦 ↔ 𝑧𝑅𝑤))
7	6	anbi2d 642	. . . . 5 ⊢ (𝑦 = 𝑤 → ((𝑧 ∈ 𝐴 ∧ 𝑧𝑅𝑦) ↔ (𝑧 ∈ 𝐴 ∧ 𝑧𝑅𝑤)))
8	5, 7	opelopabg 5517	. . . 4 ⊢ ((𝑧 ∈ V ∧ 𝑤 ∈ V) → (⟨𝑧, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑦)} ↔ (𝑧 ∈ 𝐴 ∧ 𝑧𝑅𝑤)))
9	8	el2v 3457	. . 3 ⊢ (⟨𝑧, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑦)} ↔ (𝑧 ∈ 𝐴 ∧ 𝑧𝑅𝑤))
10		brinxprnres 39046	. . . 4 ⊢ (𝑤 ∈ V → (𝑧(𝑅 ∩ (𝐴 × ran (𝑅 ↾ 𝐴)))𝑤 ↔ (𝑧 ∈ 𝐴 ∧ 𝑧𝑅𝑤)))
11	10	elv 3455	. . 3 ⊢ (𝑧(𝑅 ∩ (𝐴 × ran (𝑅 ↾ 𝐴)))𝑤 ↔ (𝑧 ∈ 𝐴 ∧ 𝑧𝑅𝑤))
12		df-br 5104	. . 3 ⊢ (𝑧(𝑅 ∩ (𝐴 × ran (𝑅 ↾ 𝐴)))𝑤 ↔ ⟨𝑧, 𝑤⟩ ∈ (𝑅 ∩ (𝐴 × ran (𝑅 ↾ 𝐴))))
13	9, 11, 12	3bitr2ri 303	. 2 ⊢ (⟨𝑧, 𝑤⟩ ∈ (𝑅 ∩ (𝐴 × ran (𝑅 ↾ 𝐴))) ↔ ⟨𝑧, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑦)})
14	1, 2, 13	eqrelriiv 5770	1 ⊢ (𝑅 ∩ (𝐴 × ran (𝑅 ↾ 𝐴))) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑦)}

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145  Vcvv 3450   ∩ cin 3898  ⟨cop 4590   class class class wbr 5103  {copab 5167   × cxp 5653  ran crn 5656   ↾ cres 5657

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2732  ax-sep 5251  ax-pr 5398

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-br 5104  df-opab 5168  df-xp 5661  df-rel 5662  df-cnv 5663  df-dm 5665  df-rn 5666  df-res 5667

This theorem is used by:  dfres4  39048