Theorem opabresex0d 48050

Description: A collection of ordered pairs, the class of all possible second components being a set, with a restriction of a binary relation is a set. (Contributed by Alexander van der Vekens, 1-Nov-2017.) (Revised by AV, 1-Jan-2021.)

Hypotheses

Ref	Expression
opabresex0d.x	⊢ ((𝜑 ∧ 𝑥𝑅𝑦) → 𝑥 ∈ 𝐶)
opabresex0d.t	⊢ ((𝜑 ∧ 𝑥𝑅𝑦) → 𝜃)
opabresex0d.y	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → {𝑦 ∣ 𝜃} ∈ 𝑉)
opabresex0d.c	⊢ (𝜑 → 𝐶 ∈ 𝑊)

Assertion

Ref	Expression
opabresex0d	⊢ (𝜑 → {⟨𝑥, 𝑦⟩ ∣ (𝑥𝑅𝑦 ∧ 𝜓)} ∈ V)

Distinct variable groups:   𝑥,𝐶,𝑦   𝜑,𝑥,𝑦

Allowed substitution hints:   𝜓(𝑥, 𝑦)   𝜃(𝑥, 𝑦)   𝑅(𝑥, 𝑦)   𝑉(𝑥, 𝑦)   𝑊(𝑥, 𝑦)

Proof of Theorem opabresex0d

Step	Hyp	Ref	Expression
1		opabresex0d.x	. . . . 5 ⊢ ((𝜑 ∧ 𝑥𝑅𝑦) → 𝑥 ∈ 𝐶)
2		opabresex0d.t	. . . . 5 ⊢ ((𝜑 ∧ 𝑥𝑅𝑦) → 𝜃)
3	1, 2	jca 520	. . . 4 ⊢ ((𝜑 ∧ 𝑥𝑅𝑦) → (𝑥 ∈ 𝐶 ∧ 𝜃))
4	3	ex 417	. . 3 ⊢ (𝜑 → (𝑥𝑅𝑦 → (𝑥 ∈ 𝐶 ∧ 𝜃)))
5	4	alrimivv 1957	. 2 ⊢ (𝜑 → ∀𝑥∀𝑦(𝑥𝑅𝑦 → (𝑥 ∈ 𝐶 ∧ 𝜃)))
6		opabresex0d.c	. . 3 ⊢ (𝜑 → 𝐶 ∈ 𝑊)
7		opabresex0d.y	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → {𝑦 ∣ 𝜃} ∈ 𝑉)
8	7	elexd 3477	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → {𝑦 ∣ 𝜃} ∈ V)
9	6, 8	opabex3d 7960	. 2 ⊢ (𝜑 → {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐶 ∧ 𝜃)} ∈ V)
10		opabbrex 7465	. 2 ⊢ ((∀𝑥∀𝑦(𝑥𝑅𝑦 → (𝑥 ∈ 𝐶 ∧ 𝜃)) ∧ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐶 ∧ 𝜃)} ∈ V) → {⟨𝑥, 𝑦⟩ ∣ (𝑥𝑅𝑦 ∧ 𝜓)} ∈ V)
11	5, 9, 10	syl2anc 595	1 ⊢ (𝜑 → {⟨𝑥, 𝑦⟩ ∣ (𝑥𝑅𝑦 ∧ 𝜓)} ∈ V)

This proof depends on syntax axioms:   → wi 4   ∧ wa 400  ∀wal 1567   ∈ wcel 2142  {cab 2740  Vcvv 3454   class class class wbr 5108  {copab 5172

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-rep 5237  ax-sep 5256  ax-pow 5335  ax-pr 5403  ax-un 7734

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-nf 1813  df-sb 2096  df-mo 2566  df-clab 2741  df-cleq 2754  df-clel 2837  df-ral 3079  df-rex 3089  df-rab 3416  df-v 3456  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-pw 4563  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-iun 4957  df-opab 5173  df-xp 5666  df-rel 5667

This theorem is used by:  opabbrfex0d  48051  opabresexd  48052