Theorem opabresex0d 47749

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 516	. . . 4 ⊢ ((𝜑 ∧ 𝑥𝑅𝑦) → (𝑥 ∈ 𝐶 ∧ 𝜃))
4	3	ex 413	. . 3 ⊢ (𝜑 → (𝑥𝑅𝑦 → (𝑥 ∈ 𝐶 ∧ 𝜃)))
5	4	alrimivv 1935	. 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 3456	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → {𝑦 ∣ 𝜃} ∈ V)
9	6, 8	opabex3d 7914	. 2 ⊢ (𝜑 → {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐶 ∧ 𝜃)} ∈ V)
10		opabbrex 7416	. 2 ⊢ ((∀𝑥∀𝑦(𝑥𝑅𝑦 → (𝑥 ∈ 𝐶 ∧ 𝜃)) ∧ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐶 ∧ 𝜃)} ∈ V) → {⟨𝑥, 𝑦⟩ ∣ (𝑥𝑅𝑦 ∧ 𝜓)} ∈ V)
11	5, 9, 10	syl2anc 590	1 ⊢ (𝜑 → {⟨𝑥, 𝑦⟩ ∣ (𝑥𝑅𝑦 ∧ 𝜓)} ∈ V)

Syntax hints:   → wi 4   ∧ wa 396  ∀wal 1545   ∈ wcel 2119  {cab 2718  Vcvv 3432   class class class wbr 5079  {copab 5141

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712  ax-rep 5206  ax-sep 5225  ax-pow 5301  ax-pr 5369  ax-un 7685

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-clab 2719  df-cleq 2732  df-clel 2815  df-ral 3055  df-rex 3065  df-rab 3393  df-v 3434  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4269  df-if 4462  df-pw 4538  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-iun 4930  df-opab 5142  df-xp 5631  df-rel 5632

This theorem is referenced by:  opabbrfex0d  47750  opabresexd  47751