Theorem bj-imdirval2lem 37158

Description: Lemma for bj-imdirval2 37159 and bj-iminvval2 37170. (Contributed by BJ, 23-May-2024.)

Hypotheses

Ref	Expression
bj-imdirval2lem.exa	⊢ (𝜑 → 𝐴 ∈ 𝑈)
bj-imdirval2lem.exb	⊢ (𝜑 → 𝐵 ∈ 𝑉)

Assertion

Ref	Expression
bj-imdirval2lem	⊢ (𝜑 → {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐵) ∧ 𝜓)} ∈ V)

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

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

Proof of Theorem bj-imdirval2lem

Step	Hyp	Ref	Expression
1		bj-imdirval2lem.exa	. . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑈)
2	1	pwexd 5321	. . 3 ⊢ (𝜑 → 𝒫 𝐴 ∈ V)
3		bj-imdirval2lem.exb	. . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑉)
4	3	pwexd 5321	. . 3 ⊢ (𝜑 → 𝒫 𝐵 ∈ V)
5		simprl 770	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐵)) → 𝑥 ⊆ 𝐴)
6		velpw 4558	. . . 4 ⊢ (𝑥 ∈ 𝒫 𝐴 ↔ 𝑥 ⊆ 𝐴)
7	5, 6	sylibr 234	. . 3 ⊢ ((𝜑 ∧ (𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐵)) → 𝑥 ∈ 𝒫 𝐴)
8		simprr 772	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐵)) → 𝑦 ⊆ 𝐵)
9		velpw 4558	. . . 4 ⊢ (𝑦 ∈ 𝒫 𝐵 ↔ 𝑦 ⊆ 𝐵)
10	8, 9	sylibr 234	. . 3 ⊢ ((𝜑 ∧ (𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐵)) → 𝑦 ∈ 𝒫 𝐵)
11	2, 4, 7, 10	opabex2 7999	. 2 ⊢ (𝜑 → {⟨𝑥, 𝑦⟩ ∣ (𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐵)} ∈ V)
12		simpl 482	. . . 4 ⊢ (((𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐵) ∧ 𝜓) → (𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐵))
13	12	ssopab2i 5497	. . 3 ⊢ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐵) ∧ 𝜓)} ⊆ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐵)}
14	13	a1i 11	. 2 ⊢ (𝜑 → {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐵) ∧ 𝜓)} ⊆ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐵)})
15	11, 14	ssexd 5266	1 ⊢ (𝜑 → {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐵) ∧ 𝜓)} ∈ V)

Syntax hints:   → wi 4   ∧ wa 395   ∈ wcel 2109  Vcvv 3438   ⊆ wss 3905  𝒫 cpw 4553  {copab 5157

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7675

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ral 3045  df-rex 3054  df-rab 3397  df-v 3440  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-opab 5158  df-xp 5629  df-rel 5630

This theorem is referenced by:  bj-imdirval2  37159  bj-iminvval2  37170