Theorem bj-dfres2 38185

Description: Alternate definition of the restriction of a class to another, as an ordered pair class abstraction: the restriction of a class to another is the class of ordered pairs whose first component is in the restricting class and whose components are related by the restricted class. (Contributed by Mario Carneiro, 5-Nov-2013.) Do not depend on df-rel 5665. (Revised by BJ, 16-Oct-2026.) This is an alternate proof of dfres2 6051; it is slightly shorter but not necessarily a gain (it uses bj-elres 38184 / elres 6027, which uses bj-elinxp 38177 / elinxp 6026): dfres2 6051 uses that both classes are relations (relres 6012 and relopab 5812) to be able to use ordered-pair extensionality eqrelriiv 5775: the latter's proof is long (considering the proofs of its parents up to ssrel 5767), but its usefulness makes it worth it.

Assertion

Ref	Expression
bj-dfres2	⊢ (𝑅 ↾ 𝐴) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑦)}

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

Proof of Theorem bj-dfres2

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-rex 3091	. . . 4 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝑥𝑅𝑦) ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ ∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝑥𝑅𝑦)))
2		an12 658	. . . . . . 7 ⊢ ((𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑦)) ↔ (𝑥 ∈ 𝐴 ∧ (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝑥𝑅𝑦)))
3	2	exbii 1881	. . . . . 6 ⊢ (∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑦)) ↔ ∃𝑦(𝑥 ∈ 𝐴 ∧ (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝑥𝑅𝑦)))
4		19.42v 1988	. . . . . 6 ⊢ (∃𝑦(𝑥 ∈ 𝐴 ∧ (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝑥𝑅𝑦)) ↔ (𝑥 ∈ 𝐴 ∧ ∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝑥𝑅𝑦)))
5	3, 4	bitr2i 279	. . . . 5 ⊢ ((𝑥 ∈ 𝐴 ∧ ∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝑥𝑅𝑦)) ↔ ∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑦)))
6	5	exbii 1881	. . . 4 ⊢ (∃𝑥(𝑥 ∈ 𝐴 ∧ ∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝑥𝑅𝑦)) ↔ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑦)))
7	1, 6	bitri 278	. . 3 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝑥𝑅𝑦) ↔ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑦)))
8		bj-elres 38184	. . 3 ⊢ (𝑧 ∈ (𝑅 ↾ 𝐴) ↔ ∃𝑥 ∈ 𝐴 ∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝑥𝑅𝑦))
9		elopab 5508	. . 3 ⊢ (𝑧 ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑦)} ↔ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑦)))
10	7, 8, 9	3bitr4i 306	. 2 ⊢ (𝑧 ∈ (𝑅 ↾ 𝐴) ↔ 𝑧 ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑦)})
11	10	eqriv 2760	1 ⊢ (𝑅 ↾ 𝐴) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑦)}

This proof depends on syntax axioms:   ∧ wa 401   = wceq 1570  ∃wex 1812   ∈ wcel 2147  ∃wrex 3090  ⟨cop 4840   class class class wbr 5108  {copab 5173   ↾ cres 5660

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735  ax-sep 5255  ax-pr 5398

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-in 4094  df-un 4154  df-sn 4589  df-pr 4591  df-op 4841  df-br 5109  df-opab 5174  df-xp 5664  df-res 5670

This theorem is used by:  relpart  38186