Theorem opabex2 7999

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 7694	. 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 5665	. 2 ⊢ (𝜑 → {⟨𝑥, 𝑦⟩ ∣ 𝜓} ⊆ (𝐴 × 𝐵))
7	3, 6	ssexd 5252	1 ⊢ (𝜑 → {⟨𝑥, 𝑦⟩ ∣ 𝜓} ∈ V)

Syntax hints:   → wi 4   ∧ wa 396   ∈ wcel 2119  Vcvv 3431  {copab 5134   × cxp 5616

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-ext 2711  ax-sep 5218  ax-pow 5294  ax-pr 5362  ax-un 7678

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-sb 2074  df-clab 2718  df-cleq 2731  df-clel 2814  df-ral 3054  df-rex 3064  df-rab 3392  df-v 3433  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-opab 5135  df-xp 5624  df-rel 5625

This theorem is referenced by:  tgjustf  28559  legval  28670  mgcoval  33065  satf00  35602  bj-imdirval2lem  37542  rfovcnvfvd  44451  sprsymrelfvlem  47965