Theorem bj-imdirval2lem 36553

Description: Lemma for bj-imdirval2 36554 and bj-iminvval2 36565. (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 5367	. . 3 ⊢ (𝜑 → 𝒫 𝐴 ∈ V)
3		bj-imdirval2lem.exb	. . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑉)
4	3	pwexd 5367	. . 3 ⊢ (𝜑 → 𝒫 𝐵 ∈ V)
5		simprl 768	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐵)) → 𝑥 ⊆ 𝐴)
6		velpw 4599	. . . 4 ⊢ (𝑥 ∈ 𝒫 𝐴 ↔ 𝑥 ⊆ 𝐴)
7	5, 6	sylibr 233	. . 3 ⊢ ((𝜑 ∧ (𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐵)) → 𝑥 ∈ 𝒫 𝐴)
8		simprr 770	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐵)) → 𝑦 ⊆ 𝐵)
9		velpw 4599	. . . 4 ⊢ (𝑦 ∈ 𝒫 𝐵 ↔ 𝑦 ⊆ 𝐵)
10	8, 9	sylibr 233	. . 3 ⊢ ((𝜑 ∧ (𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐵)) → 𝑦 ∈ 𝒫 𝐵)
11	2, 4, 7, 10	opabex2 8036	. 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 5540	. . 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 5314	1 ⊢ (𝜑 → {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐵) ∧ 𝜓)} ∈ V)

Syntax hints:   → wi 4   ∧ wa 395   ∈ wcel 2098  Vcvv 3466   ⊆ wss 3940  𝒫 cpw 4594  {copab 5200

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2695  ax-sep 5289  ax-nul 5296  ax-pow 5353  ax-pr 5417  ax-un 7718

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4315  df-if 4521  df-pw 4596  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-opab 5201  df-xp 5672  df-rel 5673

This theorem is referenced by:  bj-imdirval2  36554  bj-iminvval2  36565