Theorem opabex2 8098

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 7786	. 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 5747	. 2 ⊢ (𝜑 → {⟨𝑥, 𝑦⟩ ∣ 𝜓} ⊆ (𝐴 × 𝐵))
7	3, 6	ssexd 5342	1 ⊢ (𝜑 → {⟨𝑥, 𝑦⟩ ∣ 𝜓} ∈ V)

Syntax hints:   → wi 4   ∧ wa 395   ∈ wcel 2108  Vcvv 3488  {copab 5228   × cxp 5698

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-opab 5229  df-xp 5706  df-rel 5707

This theorem is referenced by:  tgjustf  28499  legval  28610  wksvOLD  29656  mgcoval  32959  satf00  35342  bj-imdirval2lem  37148  rfovcnvfvd  43969  sprsymrelfvlem  47364