Theorem opabex2 7754

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 7473	. 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 5788	. 2 ⊢ (𝜑 → {⟨𝑥, 𝑦⟩ ∣ 𝜓} ⊆ (𝐴 × 𝐵))
7	3, 6	ssexd 5227	1 ⊢ (𝜑 → {⟨𝑥, 𝑦⟩ ∣ 𝜓} ∈ V)

Syntax hints:   → wi 4   ∧ wa 398   ∈ wcel 2110  Vcvv 3494  {copab 5127   × cxp 5552

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5202  ax-nul 5209  ax-pow 5265  ax-pr 5329  ax-un 7460

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4567  df-pr 4569  df-op 4573  df-uni 4838  df-opab 5128  df-xp 5560  df-rel 5561

This theorem is referenced by:  tgjustf  26258  legval  26369  wksv  27400  satf00  32621  bj-imdirval2  34472  rfovcnvfvd  40351  sprsymrelfvlem  43651