Theorem opabresex0d 44777

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 512	. . . 4 ⊢ ((𝜑 ∧ 𝑥𝑅𝑦) → (𝑥 ∈ 𝐶 ∧ 𝜃))
4	3	ex 413	. . 3 ⊢ (𝜑 → (𝑥𝑅𝑦 → (𝑥 ∈ 𝐶 ∧ 𝜃)))
5	4	alrimivv 1931	. 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 3452	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → {𝑦 ∣ 𝜃} ∈ V)
9	6, 8	opabex3d 7808	. 2 ⊢ (𝜑 → {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐶 ∧ 𝜃)} ∈ V)
10		opabbrex 7326	. 2 ⊢ ((∀𝑥∀𝑦(𝑥𝑅𝑦 → (𝑥 ∈ 𝐶 ∧ 𝜃)) ∧ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐶 ∧ 𝜃)} ∈ V) → {⟨𝑥, 𝑦⟩ ∣ (𝑥𝑅𝑦 ∧ 𝜓)} ∈ V)
11	5, 9, 10	syl2anc 584	1 ⊢ (𝜑 → {⟨𝑥, 𝑦⟩ ∣ (𝑥𝑅𝑦 ∧ 𝜓)} ∈ V)

Syntax hints:   → wi 4   ∧ wa 396  ∀wal 1537   ∈ wcel 2106  {cab 2715  Vcvv 3432   class class class wbr 5074  {copab 5136

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-opab 5137  df-xp 5595  df-rel 5596

This theorem is referenced by:  opabbrfex0d  44778  opabresexd  44779