Theorem opabex2 8060

Description: Condition for an operation to be a set. (Contributed by Thierry Arnoux, 25-Jun-2019.)

Hypotheses

Ref	Expression
opabex2.1	⊢ (𝜑 → 𝐴 ∈ 𝑉)
opabex2.2	⊢ (𝜑 → 𝐵 ∈ 𝑊)
opabex2.3	⊢ ((𝜑 ∧ 𝜓) → 𝑥 ∈ 𝐴)
opabex2.4	⊢ ((𝜑 ∧ 𝜓) → 𝑦 ∈ 𝐵)

Assertion

Ref	Expression
opabex2	⊢ (𝜑 → {⟨𝑥, 𝑦⟩ ∣ 𝜓} ∈ V)

Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦   𝜑,𝑥,𝑦

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

Proof of Theorem opabex2

Step	Hyp	Ref	Expression
1		opabex2.1	. . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
2		opabex2.2	. . 3 ⊢ (𝜑 → 𝐵 ∈ 𝑊)
3	1, 2	xpexd 7756	. 2 ⊢ (𝜑 → (𝐴 × 𝐵) ∈ V)
4		opabex2.3	. . 3 ⊢ ((𝜑 ∧ 𝜓) → 𝑥 ∈ 𝐴)
5		opabex2.4	. . 3 ⊢ ((𝜑 ∧ 𝜓) → 𝑦 ∈ 𝐵)
6	4, 5	opabssxpd 5710	. 2 ⊢ (𝜑 → {⟨𝑥, 𝑦⟩ ∣ 𝜓} ⊆ (𝐴 × 𝐵))
7	3, 6	ssexd 5297	1 ⊢ (𝜑 → {⟨𝑥, 𝑦⟩ ∣ 𝜓} ∈ V)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∈ wcel 2146  Vcvv 3457  {copab 5175   × cxp 5661

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 2148  ax-9 2156  ax-ext 2737  ax-sep 5259  ax-pow 5338  ax-pr 5406  ax-un 7742

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 2744  df-cleq 2757  df-clel 2840  df-ral 3082  df-rex 3092  df-rab 3419  df-v 3459  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-opab 5176  df-xp 5669  df-rel 5670

This theorem is used by:  tgjustf  28793  legval  28904  brprlng  29243  mgcoval  33370  satf00  35903  bj-imdirval2lem  37883  rfovcnvfvd  44791  sprsymrelfvlem  48297