Theorem opabresex2 7470

Description: Restrictions of a collection of ordered pairs of related elements are sets. (Contributed by Alexander van der Vekens, 1-Nov-2017.) (Revised by AV, 15-Jan-2021.) Add disjoint variable conditions betweem 𝑊, 𝐺 and 𝑥, 𝑦 to remove hypotheses. (Revised by SN, 13-Dec-2024.)

Assertion

Ref	Expression
opabresex2	⊢ {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝑊‘𝐺)𝑦 ∧ 𝜃)} ∈ V

Distinct variable groups:   𝑥,𝑊   𝑦,𝑊   𝑥,𝐺   𝑦,𝐺

Allowed substitution hints:   𝜃(𝑥, 𝑦)

Proof of Theorem opabresex2

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fvex 6895	. 2 ⊢ (𝑊‘𝐺) ∈ V
2		elopabran 5544	. . 3 ⊢ (𝑧 ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝑊‘𝐺)𝑦 ∧ 𝜃)} → 𝑧 ∈ (𝑊‘𝐺))
3	2	ssriv 3938	. 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝑊‘𝐺)𝑦 ∧ 𝜃)} ⊆ (𝑊‘𝐺)
4	1, 3	ssexi 5291	1 ⊢ {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝑊‘𝐺)𝑦 ∧ 𝜃)} ∈ V

This proof depends on syntax axioms:   ∧ wa 401   ∈ wcel 2145  Vcvv 3453   class class class wbr 5107  {copab 5171  ‘cfv 6537

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 2734  ax-sep 5255  ax-nul 5267

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 2741  df-cleq 2754  df-clel 2837  df-ne 2958  df-rab 3415  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-sn 4588  df-pr 4590  df-uni 4871  df-br 5108  df-opab 5172  df-iota 6493  df-fv 6545

This theorem is used by:  fvmptopab  7471  mptmpoopabbrd  8083