Theorem bj-imdirval2lem 35045

Description: Lemma for bj-imdirval2 35046 and bj-iminvval2 35057. (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 5261	. . 3 ⊢ (𝜑 → 𝒫 𝐴 ∈ V)
3		bj-imdirval2lem.exb	. . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑉)
4	3	pwexd 5261	. . 3 ⊢ (𝜑 → 𝒫 𝐵 ∈ V)
5		simprl 771	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐵)) → 𝑥 ⊆ 𝐴)
6		velpw 4508	. . . 4 ⊢ (𝑥 ∈ 𝒫 𝐴 ↔ 𝑥 ⊆ 𝐴)
7	5, 6	sylibr 237	. . 3 ⊢ ((𝜑 ∧ (𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐵)) → 𝑥 ∈ 𝒫 𝐴)
8		simprr 773	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐵)) → 𝑦 ⊆ 𝐵)
9		velpw 4508	. . . 4 ⊢ (𝑦 ∈ 𝒫 𝐵 ↔ 𝑦 ⊆ 𝐵)
10	8, 9	sylibr 237	. . 3 ⊢ ((𝜑 ∧ (𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐵)) → 𝑦 ∈ 𝒫 𝐵)
11	2, 4, 7, 10	opabex2 7816	. 2 ⊢ (𝜑 → {⟨𝑥, 𝑦⟩ ∣ (𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐵)} ∈ V)
12		simpl 486	. . . 4 ⊢ (((𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐵) ∧ 𝜓) → (𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐵))
13	12	ssopab2i 5420	. . 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 5206	1 ⊢ (𝜑 → {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐵) ∧ 𝜓)} ∈ V)

Syntax hints:   → wi 4   ∧ wa 399   ∈ wcel 2110  Vcvv 3401   ⊆ wss 3857  𝒫 cpw 4503  {copab 5105

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2706  ax-sep 5181  ax-nul 5188  ax-pow 5247  ax-pr 5311  ax-un 7512

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-clab 2713  df-cleq 2726  df-clel 2812  df-nfc 2882  df-ral 3059  df-rex 3060  df-rab 3063  df-v 3403  df-dif 3860  df-un 3862  df-in 3864  df-ss 3874  df-nul 4228  df-if 4430  df-pw 4505  df-sn 4532  df-pr 4534  df-op 4538  df-uni 4810  df-opab 5106  df-xp 5546  df-rel 5547

This theorem is referenced by:  bj-imdirval2  35046  bj-iminvval2  35057