Theorem opabex2 8042

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 7737	. 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 5723	. 2 ⊢ (𝜑 → {⟨𝑥, 𝑦⟩ ∣ 𝜓} ⊆ (𝐴 × 𝐵))
7	3, 6	ssexd 5324	1 ⊢ (𝜑 → {⟨𝑥, 𝑦⟩ ∣ 𝜓} ∈ V)

Syntax hints:   → wi 4   ∧ wa 396   ∈ wcel 2106  Vcvv 3474  {copab 5210   × cxp 5674

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7724

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-opab 5211  df-xp 5682  df-rel 5683

This theorem is referenced by:  tgjustf  27721  legval  27832  wksvOLD  28874  mgcoval  32151  satf00  34360  bj-imdirval2lem  36058  rfovcnvfvd  42748  sprsymrelfvlem  46148